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MATHEMATICS 




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A NEW ERA OF THOUGHT. 



SCIENTIFIC ROMANCES. 

By C. Howard Hinton, M.A. 
Crown 8vo, cloth gilt, 6s. ; or separately, is. each. 

I. What is the Fourth Dimension? is. 

GHOSTS EXPLAINED. 

*' A short treatise of admirable clearness. . . . Mr. Hinton brings u.s, 
panting but delighted, to at least a momentary faith in the Fourth Di- 
mension, and upon the eye of this faith there opens a vista of interesting 
problems. . . . His pamphlet exhibits a boldness of speculation, and a 
power of conceiving and expressing even the inconceivable, which rouses 
one's faculties like a tonic." — Pali Mall. 

2. The Persian King; or. The Law of the Valley, is. 

THE MYSTERY OF PLEASURE AND PAIN. 

"A very suggestive and well-written speculation, by the inheritor of an 
honoured name." — Mind, 

" Will arrest the attention of the reader at once." — Knowledge, 

3. A Plane World, i.f. 

4. A Picture of our Universe, is. 

5. Casting out the Self, is. 

SECOND SERIES. 

I. On the Education of the Imagination. 

2. Many Dimensions, is. 



LONDON: SWAN SONNENSCHEIN & CO. 



A New Era of Thought. 



CHARLES HOWARD HINTON, M.A., OxoN. 

Attthor of '^ What is the Fourth Dime7isimi,'' atid other ^^ Scientific Rojuances' 




3f ffnlron : 

SWAN SONNENSCHEIN & CO., 

PATERNOSTER SQUARE. 



THE SELWOOD PRINTING WORKS, 
FEOME. and LONDON. 



PREFACE 



The MSS. which formed the basis of this book were 
committed to us by the author, on his leaving England 
for a distant foreign appointment. It was his wish that 
we should construct upon them a much more complete 
treatise than we have effected, and with that intention 
he asked us to make any changes or additions we thought 
desirable. But long alliance with him in this work has 
convinced us that his thought (especially that of a general 
philosophical character) loses much of its force if sub- 
jected to any extraneous touch. 

This feeling has induced us to print Part I. almost 
exactly as it came from his hands, although it would 
probably have received much rearrangement if he could 
have watched it through the press himself. 

Part 1 1, has been written from a hurried sketch, which 
he considered very inadequate, and which we have con- 
sequently corrected and supplemented. Chapter XI. of 
this part has been entirely re-written by us, and has thus 
not had the advantage of his supervision. This remark 
also applies to Appendix E, which is an elaboration of 
a theorem he suggested. Appendix H, and all the 
exercises have, in accordance with his wish, been written 



vi Preface. 

solely by us. It will be apparent to the reader that 
Appendix H is little more than a brief introduction to 
a very large subj.ect, which, being concerned with tes- 
saracts and solids, is really beyond treatment in writing 
and diagrams. 

This difficulty recalls us to the one great fact, upon 
which we feel bound to insist, that the matter of this 
book viust receive objective treatment from the reader, 
who will find it quite useless even to attempt to appre- 
hend it without actually building in squares and cubes 
all the facts of space which we ask him to impress on 
his consciousness. Indeed, we consider that printing, 
as a method of spreading space-knowledge, is but a "pis 
aller," and we would go back to that ancient and more 
fruitful method of the Greek geometers, and, while 
describing figures on the sand, or piling up pebbles in 
series, would communicate to others that spirit of learn- 
ing and generalization begotten in our consciousness by 
continuous contact with facts, and only by continuous 
contact with facts vitally maintained. 

ALICIA BOOLE, 
H. JOHN FALK. 

N.B. Models. — It is unquestionably a most important 
part of the process of learning space to construct these, 
and the reader should do so, however roughly and 
hastily. But, if Models are required as patterns, they 
may be ordered from Messrs. Swan Sonnenschein & Co., 



Preface. vii 

Paternoster Square, London, and will be supplied as 
soon as possible, the uncertainty as to demand for 
same not allowing us to have a large number made in 
advance. Much of the work can be done with plain 
cubes by using names without colours, but further on 
the reader will find colours necessary to enable him to 
grasp and retain the complex series of observations. 
Coloured models can easily be made by covering Kin- 
dergarten cubes with white paper and painting them 
with water-colour, and, if permanence be desired, dip- 
ping them in size and copal varnish. 



TABLE OF CONTENTS. 



PART I. 

PAGE 

Introduction . . -1-7 

CHAPTER 1. 
Scepticism and Science. Beginning of Knowledge . 8-13 

CHAPTER II. 

Apprehension of Nature. Intelligence. Study of Arrange- 
ment or Shape . 14-20 

CHAPTER III. 
The Elements of Knowledge . . . . 21-23 

CHAPTER IV. 
Theory and Practice .... . . 24-28 

CHAPTER V. 
Knowledge: Self-Elements ... . . 29-34 

CHAPTER VI. 

Function of Mind. Space against Metaphysics. Self- 
Limitation and its Test. A Plane World . . 35-4^ 



X Contents. 

PAGE 

CHAPTER VII. 
Self Elements in our Consciousness .... 47-50 

CHAPTER VIII. 
Relation of Lower to Higher Space. Theory of the .^Ether 51-60 

CHAPTER IX. 
Another View of the ^ther. Material and /Etherial Bodies 61-66 

CHAPTER X. 

Higher Space and Higher Being. Perception and In- 
spiration 67-84 

CHAPTER XI. 
Space the Scientific Basis of Altruism and Religion . . 85-99 



PART II. 

CHAPTER I. 

Three-space. Genesis of a Cube. Appearances of a Cube 

to a Plane-being 101-112 

CHAPTER II. 
Further Appearances of a Cube to a Plane-being . . 1 13-117 

CHAPTER III. 

Four-space. Genesis of a Tessaract ; its Representation 

in Three-space 118-129 

CHAPTER IV. 

Tessaract moving through Three-space. Models of the 

Sections 130-134 



Contents. xi 



PAGE 



CHAPTER V. 
Representation of Three-space by Names and in a Plane 135-148 

CHAPTER VI. 

The Means by which a Plane-being would Acquire a Con- 
ception of our Figures . .• . . . 149-155 

CHAPTER Vn. 
Four-space : its Representation in Three-space . . 156-166 

CHAPTER VIII. 
Representation of Four-space by Name. Study of Tessaracts 167-1 76 

CHAPTER IX. 
Further Study of Tessaracts . . • I77~i79 

CHAPTER X. 
Cyclical Projections ... ... 180-183 

CHAPTER XI. 
A Tessaractic Figure and its Projections . • • 184-194 

' APPENDICES. 

A. 100 Names used for Plane Space . ... 197 

B. 216 Names used for Cubic Space . . .198 

C. 256 Names used for Tessaractic Space . . 200-201 

D. List of Colours, Names, and Symbols . . 202-203 

E. A Theorem in Four-space . . 204-205 

F. Exercises on Shapes of Three Dimensions . . 205-207 

G. Exercises on Shapes of Four Dimensions . 207-209 
H. Sections of the Tessaract . ... . 209-217 
K. Drawings of the Cubic Sides and Sections of the Tes- 
saract (Models 1-12) with Colours and Names . 219-241 



INTRODUCTORY NOTE TO PART I. 



At the completion of a work, or at the completion of the first part 
of a work, the feelings are necessarily very different from those 
with which the work was begun ; and the meaning and value of the 
work itself beaf a very different appearance. It will therefore be 
the simplest and shortest plan, if I tell the reader briefly what the 
work is to which these pages'are a guide, and what I consider to 
be its value when done. ^ 

The task was to obtain a sense of the properties of higher space, 
or space of four dimensions, in the same way as that by which we 
reach a sense of our ordinary three-dimensional space. I now prefer 
to call the task that of obtaining a familiarity with higher matter, 
which, shall be as intuitive to the mind as that of ordinary matter 
has become. The expression " higher matter " is preferable to 
" higher space," because it is a somewhat hasty proceeding to split 
this concrete matter, which we touch and feel, into the abstractions 
of extension and impenetrability. It seems to me that I cannot 
think of space without matter, and therefore, as no necessity com- 
pels me to such a course, I do not split up the concrete object into 
subtleties, but I simply ask : " What is that which is to a cube or 
block or shape of any kind as the cube is to a square ? " 

In entering upon this inquiry we find the task is twofold. 
Firstly, there is the theoretical part, which is easy, viz. to set 
clearly before us the relative conditions which would obtain if 
there were a matter physically higher than this matter of ours, and 



xiv Introductory Note to Part I. 

to choose the best means of liberating our minds from the limita- 
tions imposed on it by the particular conditions under which we 
are placed. The second part of the task is somewhat laborious, 
and consists of a constant presentation to the senses of those ap- 
pearances which portions of higher matter would present, and of 
a continual dwelling on them, until the higher matter becomes 
familiar. 

The reader must undertake this task, if he accepts it at all, as an 
experiment. Those of us who have done it, are satisfied that there 
is that in the results of the experiment which make it well worthy 
of a trial. 

And in a few words I may state the general bearings of this 
work, for every branch of work has its general bearings. It is an 
attempt, in the most elementary and simple domain, to pass from 
the lower to the higher. In pursuing it the mind passes from one 
kind of intuition to a higher one, and with that transition the 
horizon of thought is altered. It becomes clear that there is a 
physical existence transcending the ordinary physical existence ; 
and one becomes inclined to think that the right direction to look 
is, not away from matter to spiritual existences, but towards the 
discovery of conceptions of higher matter, and thereby of. those 
material existences whose definite relations to us are apprehended 
as spiritual intuitions. Thus, " material " would simply mean 
" grasped by the intellect, become known and familiar.'' Our ap- 
prehension of anything which is not expressed in terms of matter, 
is vague and indefinite. To realize and live with that which we 
vaguely discern, we need to apply the intuition of higher matter to 
the world around us. And this seems to me the great inducement 
to this study. Let us form our intuition of higher space, and then 
look out upon the world. 

Secondly, in this progress from ordinary to higher matter, as a 
general type of progress from lower to higher, we make the fol- 
lowing observations. Firstly, we become aware that there are 



Introductory Note to Part I. xv 

certain limitations affecting our regard. Secondly, we discover by 
our reason what those limitations are, and then force ourselves to 
go through the experience which would be ours if the limitations 
did not affect us. Thirdly, we become aware of a capacity within 
us for transcending those limitations, and for living in the higher 
mode as we had lived in the previous one. 

We may remark that this progress from the ordinary to the 
higher kind of matter demands an absolute attention to details. 1 1 
is only in the retention of details that such progress becomes pos- 
sible. And as, in this question of matter, an absolute and uncon- 
ventional examination gives us the indication of a higher, so, 
doubtless, in other questions, if we but come to facts without pre- 
supposition, we begin to know that there is a higher and to dis- 
cover indications of the way whereby we can approach. That way 
lies in the fulness of detail rather than in the generalization. 

Biology has shown us that there is a universal order of forms 
or organisms, passing from lower to higher. Therein we find an 
indication that we ourselves take part in this progress. And in 
using the little cubes we can go through the process ourselves, and 
learn what it is in a little instance. 

But of all the ways in which the confidence gained from this 
lesson can be applied, the nearest to us lies in the suggestion it 
gives, — and more than the suggestion, if incMnation to think be 
counted for anything, — in the suggestion of that which is higher 
than ourselves. We, as individuals, are not the limit and end-all, 
but there is a higher being than ours. What our relation to it is, 
we cannot tell, for that is. unlike our relation to anything we know. 
But, perhaps all that happens to us is, could we but grasp it, our 
relation to it. 

At any rate, the discovery of it is the great object beside which 
all else is as secondary as the routine of mere existence is to 
companionship. And the method of discovery is full knowledge of 
each other. Thereby is the higher being to be known. In as much 



xvi Introductory Note to Part /. 

as the least of us knows and is known by another, in so much does 
he know the higher. Thus, scientific prayer is when two or three 
meet together, and, in the belief of one higher than themselvesi 
mutually comprehend that vision of the higher, which each one is, 
and, by absolute fulness of knowledge of the facts of each other's 
personality, strive to attain a knowledge of that which is to each of 
their personalities as a higher figure is to its solid sides. 

C. H. H. 



A NEW ERA OF THOUGHT. 



PART I. 

INTRODUCTION. 



There are no new truths in this book, but it consists 
of an effort to impress upon and bring home to the 
mind some of the more modern developments of thought. 
A few sentences of Kant, a few leading ideas of Gauss 
and Lobatschewski form the material out of which it 
is built up. 

It may be thought to be unduly long ; but it must 
be remembered that in these times there is a twofold 
process going on — one of discovery about external 
nature, one of education, by which our minds are 
brought into harmony with that which we know. In 
certain respects we find ourselves brought on by the 
general current of ideas — we feel that matter is permanent 
and cannot be annihilated, and it is almost an axiom 
in our minds that energy is persistent, and all its trans- 
formations remains the same in amount. But there are 
other directions in which there is need of definite train- 
ing if we are to enter into the thoughts of the time. 

And it seems to me that a return to Kant, the creator 
of modern philosophy, is the first condition. Now of 
Kant's enormous work only a small part is treated here, 
but with the difference that should be found between the 
work of a master and that of a follower. Kant's state- 



2 A New Era of Thought. 

ments are taken as leading ideas, suggesting a field of 
work, and it is in detail and manipulation merely that 
there is an opportunity for workmanship. 

Of Kant's work it is only his doctrine of space which 
is here experimented upon. With Kant the perception 
of things as being in space is not treated as it seems so 
obvious to do. We should naturally say that there is 
space, and there are things in it. From a comparison 
of those properties which are common to all things we 
obtain the properties of space. But Kant says that 
this property of being in space is not so much a quality 
of any definable objects, as the means by which we 
obtain an apprehension of definable objects — it is the 
condition of our mental work. 

Now as Kant's doctrine is usually commented on, the 
negative side is brought into prominence, the positive 
side is neglected. It is generally said that the mind 
cannot perceive things in themselves, but can only 
apprehend them subject to space conditions. And in 
this way the space conditions are as it were considered 
somewhat in the light of hindrances, whereby we are 
prevented from seeing what the objects in themselves 
truly are. But if we take the statement simply as it 
is — that we apprehend by means of space — then it is 
equally allowable to consider our space sense as a 
positive means by which the mind grasps its experience. 

There is in so many books in which the subject is 
treated a certain air of despondency — as if this space 
apprehension were a kind of veil which shut us off from 
nature. But there is no need to adopt this feeling. 
The first postulate of this book is a full recognition of 
the fact, that it is by means of space that we apprehend 
what is. Space is the instrument of the mind. 

And here for the purposes of our work we can avoid 
all metaphysical discussion. Very often a statement 



Introduction. 3 

which seems to be very deep and abstruse and hard 
to grasp, is simply the form into which deep thinkers 
have thrown a very simple and practical observation. 
And for the present let us look on Kant's great doctrine 
of space from a practical point of view, and it comes to 
this — it is important to develop the space sense, for it is 
the means by which we think about real things. 

There is a doctrine which found much favour with 
the first followers of Kant, that also affords us a simple 
and practical rule of work. It was considered by Fichte 
that the whole external world was simply a projection 
from the ego, and the manifold of nature was a recogni- 
tion by the spirit of itself What this comes to as a 
practical rule is, that we can only understand nature in 
virtue of our own activity ; that there is no such thing 
as mere passive observation, but every act of sight and 
thought is an activity of our own. 

Now according to Kant the space sense, or the in- 
tuition of space, is the most fundamental power of the 
mind. But I do not find anywhere a systematic and 
thoroughgoing education of the space sense. In every 
practical pursuit it is needed — in some it is developed. 
In geometry it is used ; but the great reason of failure 
in education is that, instead of a systematic training 
of the space sense, it is left to be organized by ac- 
cident and is called- upon to act without having been 
formed. According to Kant and according to common 
experience it will be found that a trained thinker is one 
in whom the space sense has been well developed. 

With regard to the education of the space sense, I 
must ask the indulgence of the reader. It will seem 
obvious to him that any real pursuit or real observation 
trains the space sense, and that it is going out of the 
way to undertake any special discipline. 

To this I would answer that, according to my own 



4 A New Era of Thought. 

experience, I was perfectly ignorant of space relations 
myself before I actually worked at the subject, and 
that directly I got a true view of space facts a whole 
series of conceptions, which before I had known merely 
by repute and grasped by an effort, became perfectly 
simple and clear to me. 

Moreover, to take one instance : in studying the 
relations of space we always have to do with coloured 
objects, we always have the sense of weight ; for if the 
things themselves have no weight, there is always a 
direction of up and down — which implies the sense of 
weight, and to get rid of these elements requires careful 
sifting. But perhaps the best point of view to take is 
this — if the reader has the space sense well developed 
he will have no difficulty in going through the part of 
the book which relates to it, and the phraseology will 
serve him for the considerations which come next. 

Amongst the followers of Kant, those who pursued 
one of the lines of thought in his works have attracted 
the most attention and have been considered as his suc- 
cessors. Fichte, Schelling, Hegel have developed cer- 
tain tendencies and have written remarkable books. 
But the true successors of Kant are Gauss and Lobat- 
chewski. 

• For if our intuition of space is the means by which we 
apprehend, then it follows that there may be different 
kinds of intuitions of space. Who can tell what the ab- 
solute space intuition is ? This intuition of space must 
be coloured, so to speak, by the conditions of the being 
which uses it. 

Now, after Kant had laid down his doctrine of space, 
it was important to investigate how much in our space 
intuition is due to experience — is a matter of the phy- 
sical circumstances of the thinking being — and how 
much is the pure act of the mind. 



Introduction. 5 

The only way to investigate this is the practical way, 
and by a remarkable analysis the great geometers above 
mentioned have shown that space is not limited as or- 
dinary experience would seem to inform us, but that we 
are quite capable of conceiving different kinds of space. 

Our space as we ordinarily think of it is conceived as 
limited — not in extent, but in a certain way which can 
only be realized when we think of our ways of measur- 
ing space objects. It is found that there are only three 
independent directions in which a body can be measured 
— it must have height, length and breadth, but it has 
no more than these dimensions. If any other measure- 
ment be taken in it, this new measurement will be found 
to be compounded of the old measurements. It is im- 
possible to find a point in the body which could not 
be arrived at by travelling in combinations of the three 
directions already taken. 

But why should space be limited to three independent 
directions \ 

Geometers have found that there is no reason why 
bodies should be thus limited. As a matter of fact all 
the bodies which we can measure are thus limited. So 
we come to this conclusion, that the space which we use 
for conceiving ordinary objects in the world is limited 
to three dimensions. But it might be possible for there 
to be beings living in a world such that they would con- 
ceive a space of four dimensions. All that we can say 
about such a supposition is, that it is not demanded by 
our experience. It may be that in the very large or 
the very minute a fourth dimension of space will have 
to be postulated to account for parts — but with regard 
to objects of ordinary magnitudes we are certainly not 
in a four dimensional world. 

And this was the point at which about ten years ago 
I took up the inquiry. 



6 A New Era of Thought. 

It is possible to say a great deal about space of higher 
dimensions than our own, and to work out analytically 
many problems which suggest themselves. But can we 
conceive four-dimensional space in the same way in 
which we can conceive our own space ? Can we think 
of a body in four dimensions as a unit having properties 
in the same way as we think of a body having a definite 
shape in the spate with which we are familiar ? 

Now this question, as every other with which I am 
acquainted, can only be answered by experiment. And 
I commenced a series of experiments to arrive at a con- 
clusion one way or the other. 

It is obvious that this is not a scientific inquiry — but 
one for the practical teacher. 

And just as in experimental researches the skilful 
manipulator will demonstrate a law of nature, the less 
skilled manipulator will fail ; so here, everything de- 
pended on the manipulation. I was not sure that this 
power lay hidden in the mind, but to put the question 
fairly would surely demand every resource of the prac- 
tical art of education. 

And so it proved to be ; for after many years of work, 
during which the conception of four-dimensional bodies 
lay absolutely dark, at length, by a certain change of 
plan, the whole subject of four-dimensional existence 
became perfectly clear and easy to impart. 

There is really no more difficulty in conceiving four- 
dimensional shapes, when we go about it the right way, 
than in conceiving the idea of solid shapes, nor is there 
any mystery at all about it. 

When the faculty is acquired — or rather when it is 
brought into consciousness, for it exists in every one in 
imperfect form — a new horizon opens. The mind ac- 
quires a development of power, and in this use of ampler 
space as a mode of thought, a path is opened by using 



Introdtiction. •/ 

that very truth which, when first stated by Kant, seemed 
to close the mind within such fast limits. Our percep- 
tion is subject to the condition of being in space. But 
space is not limited as we at first think. 

The next step after having formed this power of con- 
ception in ampler space, is to investigate nature and see 
what phenomena are to be explained by four-dimen- 
sional relations. 

But this part of the subject is hardly one for the same 
worker as the one who investigates how to think in four- 
dimensional space. The work of building up the power 
is the work of the practical educator, the work of apply- 
ing it to nature is the work of the scientific man. And 
it is not possible to accomplish both tasks at the same 
time. Consequently the crown is still to be won. Here 
the method is given of training the mind ; it will be an 
exhilarating moment when an investigator comes upon 
phenomena which show that external nature cannot be 
explained except by the assumption of a four-dimen- 
sion space. 

The thought of the past ages has used the conception 
of a three-dimensional space, and by that means has 
classified many phenomena and has obtained rules for 
dealing with matters of great practical utility. The 
path which opens immediately before us in the future 
is that of applying the conception of four-dimensional 
space to the phenomena of nature, and of investigating 
what can be found out by this new means of appre- 
hension. 

In fact, what has been passed through may be called 
the three-dimensional era ; Gauss and Lobatchewski 
have inaugurated the four-dimensional era. 



CHAPTER I. 

SCEPTICISM AND SCIENCE. BEGINNING OF 
KNOWLEDGE. 

The following pages have for their object to induce 
the reader to apply himself to the study, in the first 
place of Space, and then of Higher Space ; and, there- 
fore, I have tried by indirect means to show forth 
those thoughts and conceptions to which the practical 
work leads. 

And I feel that I have a great advantage in this 
project, inasmuch as many of the thoughts which spring 
up in the mind of one who studies higher space, and 
many of the conceptions to which he is driven, turn out 
to be nothing more nor less than old truths — the pro- 
perty of every mind that thinks and feels — truths which 
are not generally associated with the scientific appre- 
hension of the world, but which are not for that reason 
any the less valuable. 

And for my own part I cannot do more than put 
them forward in a very feeble and halting manner. 
For I have come upon them, not in the way of feeling 
or direct apprehension, but as the result of a series of 
works undertaken purely with the desire to know — a 
desire which did not lift itself to the height of expecting 
or looking for the beautiful or the good, but which 
simply asked for something to know. 

For I found myself — and many others I find do so 
also — I found myself in respect to knowledge like a 
man who is in the midst of plenty and yet who cannot 
find anything to eat. All around me were the evidences 



Scepticism and Science. 9 

of knowledge— the arts, the sciences, interesting talk, 
useful inventions — and yet I myself was profited 
nothing at all ; for somehow, amidst all this activity, I 
was left alone, I could get nothing which I could know. 

The dialect was foreign to me — its inner meaning 
was hidden. If I would, imitating the utterance of 
my fellows, say a few words, the effort was forced, the 
whole result was an artificiality, and, if successful, would 
be but a plausible imposture. 

The word " sceptical " has a certain unpleasant asso- 
ciation attached to it, for it has been used by so many 
people who are absolutely certain in a particular line, 
and attack other people's convictions. But to be scep- 
tical in the real sense is a far more unpleasant state of 
mind to the sceptic than to any one of his companions. 
For to a mind that inquires into what it really does 
know, it is hardly possible to enunciate complete sen- 
tences, much less to put before it those complex ideas 
which have so large a part in true human life. 

Every word we use has so wide and fugitive a mean- 
ing, and every expression touches or rather grazes fact 
by so very minute a point, that, if we wish to start with 
something which we do know, and thence proceed in a 
certain manner, we are forced away from the study of 
reality and driven to an artificial system, such as logic 
or mathematics, which, starting from postulates and 
axioms, develops a body of ideal truth which rather 
comes into contact with nature than is nature. 

Scientific achievement is reserved for those who are 
content to absorb into their consciousness, by any means 
and by whatever way they come, the varied appearances 
of nature, whence and in which by reflection they find 
floating as it were on the sea of the unknown, certain 
similarities, certain resemblances and analogies, by 
means of which they collect together a body of possible 



lO A Nczo Era of Thought. 

predictions and inferences ; and in nature they find 
correspondences which are actually verified. Hence 
science exists, although the conceptions in the mind 
cannot be said to have any real correspondence in 
ndture. 

We form a set of conceptions in the mind, and the 
relations between these conceptions give us relations 
which we find actually vibrating in the world around 
us. But the conceptions themselves are essentially 
artificial. 

We have a conception of atoms ; but no one supposes 
that atoms actually exist. We suppose a force varying 
inversely as the square of the distance ; but no one 
supposes such a mysterious thing to really be in nature. 
And when we come to the region of descriptive science, 
when we come to simple observation, we do not find 
ourselves any better provided with a real knowledge of 
nature. If, for instance, we think of a plant, we picture 
to ourselves a certain green shape, of a more or less 
definite character. This green shape enables us to 
recognise the plant we think of, and to describe it to a 
certain extent. But if we inquire into our imagination 
of it, we find that our mental image very soon diverges 
from the fact. If, for instance, we cut the plant in half, 
we find cells and tissues of various kinds. If we examine 
our idea of the plant, it has merely an external and 
superficial resemblance to the plant itself. It is a mental 
drawing meeting the real plant in external appearance ; 
but the two things, the plant and our thought of it, 
come as it were from different sides — they just touch 
each other as far as the colour and shape are concerned, 
but as structures and as living organisms they are as 
wide apart as possible. 

Of course by observation and study the image of a 
plant which we bear in our minds may be made to re- 



Beginning of Knowledge. 1 1 

semble a plant as found in the fields more and more. 
But the agreement with nature lies in the multitude of 
points superadded on to the notion of greenness which 
we have at first — there is no natural starting-point where 
the mind meets nature, and whence they can travel hand 
in hand. 

It almost seems as if, by sympathy and feeling, a 
human being was easier to know than the simplest ob- 
ject. To know any object, however simple, by the reason 
and observation requires an endless process of thought 
and looking, building up the first vague impression into 
something like in more and more respects. While, on 
the other hand, in dealing with human beings there is 
an inward sympathy and capacity for knowing which is 
independent of, though called into play by, the obser- 
vation of the actions and outward appearance of the 
human being. 

But for the purpose of knowing we must leave out 
these human relationships. They are an affair of in- 
stinct and inherited unconscious experience. The mind 
may some day rise to the level of these inherited appre- 
hensions, and be able to explain them ; but at present 
it is far more than overtasked to give an account of the 
simplest portions of matter, and is quite inadequate to 
give an account of the nature of a human being. 

Asking, then, what there was which I could know, 
I found no point of beginning. There were plenty of 
ways of accumulating observations, but none in which 
one could go hand in hand with nature. 

A child is provided in the early part of its life with a 
provision of food adapted for it. But it seemed that our 
minds are left without a natural subsistence, for on the 
one hand there are arid mathematics, and on the other 
there is observation, and in observation there is, out 
of the great mass of constructed, mental images, but little 



12 A New Era of Thought. 

which the mind can assimilate. To the worker at science 
of course this crude and omnivorous observation is 
everything'; but if we ask for something which we can 
know, it is like a vast mass of indigestible material 
with every here and there a fibre or thread which we 
can assimilate. 

In this perplexity I was reduced to the last condition 
of mental despair ; and in default of finding anything 
which I could understand in nature, I was sufificiently 
humbled to learn anything which seemed to afford a 
capacity of being known. 

And the objects which came before me for this en- 
deavour were the simple ones which will be plentifully 
used in the practical part of this book. For I found 
that the only assertion I could make about external 
objects, without bringing in unknown and unintel- 
ligible relations, was this : I could say how things were 
arranged. If a stone lay between two others, that was 
a definite and intelligible fact, and seemed primary. As 
a stone itself, it was an unknown somewhat which one 
could get more and more information about the more 
one studied the various sciences. But granting that 
there were some things there which we call stones, the 
way they were arranged was a simple and obvious fact 
which could be easily expressed and easily remembered. 

And so in despair of being able to obtain any other 
kind of mental possession in the way of knowledge, I 
commenced to learn arrangements, and I took as the 
objects to be arranged certain artificial objects of a 
simple shape. I built up a block of cubes, and giving 
each a name I learnt a mass of them. 

Now I do not recommend this as a thing to be done. 
All I can say is that genuinely then and now it seemed 
and seems to be the only kind of mental possession which 
one can call knowledge. It is perfectly definite and 



Beginning of Knowledge. 13 

certain. I could tell where each cube came and how 
it was related to each of the others. As to the cube it- 
self, I was profoundly ignorant of that ; but assuming 
that as a necessary starting-point, taking that as granted, 
I had a definite mass of knowledge. 

But I do not wish to say that this is better than any 
kind of knowledge which other people may find come 
home to them. All I want to do is to take this humble 
beginning of knowledge and show how inevitably, by 
devotion to it, it leads to marvellous and far-distant 
truths, and how, by a strange path, it leads directly into 
the presence of some of the highest conceptions which 
great minds have given us. 

I do not think it ought to be any objection to an in- 
quiry, that it begins with obvious and common details. 
In fact I do not think that it is possible to get anything 
simpler, with less of hypothesis about it, and more ob- 
viously a simple taking in of facts than the study of the 
arrangement of a block of cubes. 

Many philosophers have assumed a starting point 
for their thought. I want the reader to accept a very 
humble one and see what comes of it. If this leads us 
to anything, no doubt greater results will come from more 
ambitious beginnings. 

And now I feel that I have candidly exposed myself 
to the criticism of the reader. If he will have the 
patience to go on, we will begin and build up on our 
foundations. 



CHAPTER II. 

APPREHENSION OF NATURE. INTELLIGENCE. STUDY 
OF ARRANGEMENT OR SHAPE. 

Nature is that which is around us. But it is by no 
means easy to get to nature. The savage living we may 
say in the bosom of nature, is certainly unapprehensive 
of it, in fact it has needed the greatness of a Wordsworth 
and of generations of poets and painters to open our 
eyes even in a slight measure to the wonder of nature. 

Thus it is clear that it is not by mere passivity that 
we can comprehend nature ; it is the goal of an activity, 
not a free gift. 

And there are many ways of apprehending nature. 
There are the sounds and sights of nature which de- 
light the senses, and the involved harmonies and the 
secret affinities which poetry makes us feel ; then, more- 
over, there is the definite knowledge of natural facts in 
which the memory and reason are employed, 

Thus we may divide our means of coming into con- 
tact with nature into three main channels : the senses, 
the imagination, and the mind. The imagination is 
perhaps the highest faculty, but we leave it out of con- 
sideration here, and ask : How can we bring our minds 
into contact with nature ? 

Now when we see two people of diverse characters 
we sometimes say that they cannot understand one 
another — there is nothing in the one by which he can 
understand the other — he is shut out by a limitation of 
his own faculties. 



Apprehension of Nature. Intelligence. 15 

And thus our power of understanding nature depends 
on our own possession ; it is in virtue of some mental 
activity of our own that we can apprehend that outside 
activity which we call nature. And thus the training to 
enable us to approach nature with our minds will be 
some active process on our own part. 

In the course of my experience as a teacher I have 
often been struck by the want of the power of reason 
displayed by pupils ; they are not able to put two and 
two together, as the saying goes, and I have been at 
some pains to investigate wherein this curious deficiency 
lies, and how it can be supplied. And I have found 
that there is in the curriculum no direct cure for it — the 
discipline which supplies it is not one which comes into 
school methods, it is a something which most children 
obtain in the natural and unsupervised education of their 
first contact with the world, and lies before any recog- 
nised mode of distinction. They can only understand 
in virtue of an activity of their own, and they have not 
had sufficient exercise in this activity. 

In the present state of education it is impossible to 
diverge from the ordinary routine. But it is always 
possible to experiment on children who are out of the 
common line of education. And I believe I am amply 
justified by the result of my experiments. 

I have seen that the same activity which I have 
found makes that habit of mind which we call inteUi- 
gence in a child, is the source of our common and every- 
day rational intellectual work, and that just as the 
faculties of a child can be called forth by it, so also the 
powers of a man are best prepared by the same means, 
but on an ampler scale. 

A more detailed development of the practical work 
of Part II., would be the best training for the mind oi 
a child. An extension of the work of that Part would 



1 6 A New Era of Thought. 

be the training which, hand in hand with observation and 
recapitulation, would best develop a man's thought power. 

In order to tell what the activity is by the prosecution 
of which we can obtain mental contact with nature 
we should observe what it is which we say we " under- 
stand " in any phenomenon of nature which has become 
clear to us. 

When we look at a bright object it seems very dif- 
ferent from a dull one. A piece of bright steel hardly 
looks like the same substance as a piece of dull steel. 
But the difference of appearance in the two is easily 
accounted for by the different nature of the surface in 
the two cases ; in the one all the irregularities are done 
away with, and the rays of light which fall on it are sent 
off again without being dispersed and broken up. In 
the case of the dull iron the rays of light are broken up 
and divided, so that they are not transmitted with 
intensity in any one direction, but flung off in all sorts 
of directions. 

Here the difference between the bright object and the 
dull object lies in the arrangement of the particles on its 
surface and their influence on the rays of light. 

Again, with light itself the differences of colour are 
explained as being the effect on us of rays of different 
rates of vibration. Now a vibration is essentially this, a 
series of arrangements of matter which follow each 
other in a closed order, so that when the set has been 
run through, the first arrangement follows again. The 
whole theory of light is an account of arrangements 
of the particles In the transmitting medium, only the 
arrangements alter — are not permanent in any one 
characteristic, but go through a complete cycle of 
varieties. 

Again, when the movements of the heavenly bodies 
are deduced from the theory of universal gravitation, 



Study of Arrangement or Shape. 1 7 

what we primarily do is to take account of arrangement ; 
for the law of gravity connects the movements which 
the attracted bodies tend to make with their distances, 
that is, it shows how their movements depend on their 
arrangement. And if gravity as a force is to be explained 
itself, the suppositions which have been put forward 
resolve it into the effect of the movements of small 
bodies ; that is to say, gravity, if explained at all, is 
explained as the result of the arrangement and altering 
arrangements of small particles. 

Again, to take the idea which proceeding from Goethe 
casts such an influence on botanical observation. 
Goethe (and also Wolf) laid down that the parts of a 
flower were modified leaves — and traced the stages and 
intermediate states between the ordinary green leaf and 
the most gorgeous petal or stamen or carpel, so un- 
like a leaf in form and function. 

Now the essential value in this conception is, that 
it enables us to look, upon these different organs of a 
plant as modifications of one and the same organ — it 
enables us to think about the different varieties of the 
flower head as modifications of one single plant form. 
We can trace correspondences between them, and are 
led to possible explanations of their growth. And all 
this is done by getting rid of pistil and stamen as separ- 
ate entities, and looking on them as leaves, and their 
parts due to different arrangement of the leaf structure. 
We have reduced these diverse objects to a common 
element, we have found the unit by whose arrangements 
the whole is produced. And in this department of 
thought, as also to take another instance, in chemistry, 
to understand is practically this : we find units (leaves 
or atoms) combinations of which account for the results 
which we see. Thus we see that that which the mind 
essentially apprehends is arrangement. 

C 



1 8 New Era of Thought. 

And this holds over the whole range of mental work, 
from the simplest observation to the most complex theory. 
When the eye takes in the form of an external object 
there is something more than a sense impression, some- 
thing more than a sensation of greenness and light and 
dark. The mind works as well as the sense, and these 
sense impressions are definitely grouped in what we call 
the shape of the object. The essential act of perceiving 
lies in the apprehension of a shape, and a shape is an 
arrangement of parts. It does not matter what these 
parts are ; if we take meaningless dots of colour and 
arrange them we obtain a shape which represents the 
appearance of a stone or a leaf to a certain degree. If 
we want to make our representation still more like, we 
must treat each of the dots as in themselves arrange- 
ments, we must compose each of them by many strokes 
and dots of the bjr ush. But even in this case we have 
not got anything else besides arrangement. The ulti- 
mate element, the small items of light and shade or of 
colour, are in themselves meaningless ; it is in their ar- 
rangement that the likeness of the representation consists. 

Thus, from a drawing to our notion of the planetary 
system, all our contact with nature lies in this, in an 
appreciation of arrangement. 

Hence to prepare ourselves for the understanding of 
nature, we must " arrange." In virtue of our activity in 
making arrangements we prepare ourselves to do what 
is called understand nature. Or we may say, that 
which we call understanding nature is to discern some- 
thing similar in nature to that which we do when we 
arrange elements into compounded groups. 

Now if we study arrangement in the active way, we 
must have something to arrange ; and the things we 
work with may be either all alike, or each of them vary- 
ing from every other. 



Study of Arrangement or Shape. 19 

If the elements aire not alike then we are not study- 
ing pure arrangement; but our knowledge is affected by 
the compound nature of that with which we deal. If 
the elements are all alike, we have what we call units. 
Hence the discipline preparatory for the understanding 
of nature is the active arrangement of like units. 

And this is very much the case with all educational 
processes ; only the things chosen to arrange are in 
general words, which are so complicated and carry such 
a train of association that, unless the mind has already 
acquired a knowledge of arrangement, it is puzzled and 
hampered, and never gets a clear apprehension of what 
its work is. 

Now what shall we choose for our units ? Any unit 
would do ; but it ought to be a real thing — it ought to 
be something which can be touched and seen, not some- 
thing which no one has ever touched or seen, and which 
is even incapable of definition, like a " number." 

I would divide studies into two classes : those which 
create the faculty of arrangement, and those which use 
it and exercise it. Mathematics exercises it, but I do 
not think it creates it ; and unfortunately, in mathe- 
matics as it is now often taught, the pupil is launched 
into a vast system of symbols — the whole use and 
meaning of symbols (namely, as means to acquire a 
clear grasp of facts) is lost to him. 

Of the possible units which will serve, I take the 
cube ; and I have found that whenever I took any other 
unit I got wrong, puzzled and lost my way. With the 
cube one does not get along very fast, but everything 
is perfectly obvious and simple, and builds up into a 
whole of which every part is evident. 

And I must ask the reader to absolutely erase from 
his mind all desire or wish to be able to predict or 
assert anything about nature, and he must please look 



20 New Era of Thought. 

with horror on any mental process by which he gets at 
a truth in an ingenious but obscure and inexplicable 
way. Let him take nothing which is not perfectly clear, 
patent and evident, demonstrable to his senses, a simple 
repetition of obvious fact. 

Our work will then be this : a study, by means of 
cubes, of the facts of arrangement. And the process of 
learning will be an active one of actually putting up the 
cubes. In this way we do for the mind what Words- 
worth does for the imagination — we bring it into con- 
tact with nature. 



CHAPTER III. 

THE ELEMENTS OF KNOWLEDGE. 

There are two elements which enter into our know- 
ledge with respect to any phenomenon. 

If, for instance, we take the sun, and ask ourselves 
what we observe, we notice that it is a bright, moving 
body ; and of these two qualities, the brightness and 
the movement, each seems equally predicable of the 
sun. It does move, and it is bright. 

Now further study discloses to us that there is a 
difference between these two affirmations. The motion 
of the sun in its diurnal course round the earth is only 
apparent ; but it is really a bright, hot body. 

Now of these two assertions which the mind naturally 
makes about the sun, one — that it is moving — depends 
on the relation of the beholder to the sun, the other is 
true about the sun itself The observed motion depends 
on a fact affecting oneself and having nothing to do 
with the sun, while the brightness is really a quality of 
the sun itself 

Now we will call those qualities or appearances which 
we notice in a body which are due to the particular 
conditions under which oneself is placed in observing 
it, the self elements ; those facts about it which are 
independent of the observer's particular relationship we 
will call the residual element. Thus the sun's motion 
is a self element in our thought of the sun, its brightness 
is a residual element. 



2 2 New Era of Thought. 

It is not, of course, possible to draw a line distinctly 
between the self elements and the residual elements. 
For instance, some people have denied that brightness 
is a quality of things, but that it depends on the capacity 
of the being for receiving sensations ; and for brightness 
they would substitute the assertion that the sun is 
giving forth a great deal of energy in the form of heat 
and light. 

But there is no object in pursuing the discussion 
further. The main distinction is sufficiently obvious. 
And it is important to separate the self elements in- 
volved in our knowledge as far as possible, so that the 
residual elements may be kept for our closer attention. 
By getting rid of the self elements we put ourselves in 
a position in which we can propound sensible questions. 
By getting rid of the notion of its circular motion round 
the earth we prepare our way to study the sun as it 
really is. We get the subject clear of complications 
and extraneous considerations. 

It would hardly be worth while to dwell on this con- 
sideration were it not of importance in our study of 
arrangement. But the fact is that directly a subject 
has been cleared of the self elements, it seems so absurd 
to have had them introduced at all that the great diffi- 
culty there was in getting rid of them is forgotten. 

With regard to the knowledge we have at the present 
day about scientific matters, there do not seem to be 
any self elements present. But the worst about a self 
element is, that its presence is never dreamed of till it 
. is got rid of ; to know that it is there is to have done 
away with it. And thus our body of knowledge is like 
a fluid which keeps clear, not because there are no sub- 
stances in solution, but because directly they become 
evident they fall down as precipitates. 

Now one of our serious pieces of work will be to get 



The Elements of Knowledge. 23 

rid of the self elements in the knowledge of arrange- 
ment. 

And the kind of knowledge which we shall try to 
obtain will be somewhat different from the kind of 
knowledge which we have about events or natural 
phenomena. In the large subjects which generally 
occupy the mind the things thought of are so compli- 
cated that every detail cannot possibly be considered. 
The principles of the whole are realized, and then at 
any required time the principles can be worked out. 
Thus, with regard to a knowledge of the planetary 
system, it is said to be known if the law of movement 
of each of the planets is recognized, and their positions 
at any one time committed to memory. It is not our 
habit to remember their relative positions with regard 
to one another at many intervals, so as to have an 
exhaustive catalogue of them in our minds. But with 
regard to the elements of knowledge with which we 
shall work, the subject is so simple that we may justly 
demand of ourselves that we will know every detail. 

And the knowledge we shall acquire will be much 
more one of the sense and feeling than of the reason. 
We do not want to have a rule in our minds by which 
we can recall the positions of the different cubes, but 
we want to have an immediate apprehension of them. 
It was Kant who first pointed out how much of, thought 
there was embodied in the sense impressions ; and it is 
this embodied thought which we wish to form. 



CHAPTER IV. 

THEORY AND PRACTICE. 

Both in science and in morals there is an important 
distinction to be drawn between theory and practice. 
A knowledge of chemistry does not consist in the in- 
tellectual appreciation of different theories and principles, 
but in being able' to act in accordance with the facts 
of chemical combination, so that by means of the ap- 
pliances of chemistry practical results are produced. 
And so in morals — the theoretic acquaintance with the 
principles of human action may consist with a marked 
degree of ignorance of how to act amongst other human 
beings. 

Now the use of the word " learn " has been much 
restricted to merely theoretic studies. It requires to be 
enlarged to the scientific meaning. And to know, should 
include practice in actual manipulation. 

Let us take an instance. We all know what justice 
is, and any child can be taught to tell the difference 
between acting justly and acting unjustly. But it is a 
different thing to teach them to act with justice. Some- 
thing is done which affects them unpleasantly. They 
feel an impulse to retahate. In order to see what justice 
demands they have to put their personal feeling on one 
side. They have to get rid of those conditions under 
which they apprehended the effects of the action at first, 
and they have to look on it from another point of view. 
Then they have to act in accordance with this view. 



Theory and Practice. 25 

Now there are two steps — one an intellectual one of 
understanding, one a practical one of carrying out the 
view. Neither is a moral step. One demands intelli- 
gence, the other the formation of a habit, and this habit 
can be" inculcated by precept, by reward and punish- 
ment, by various means. But as human nature is 
constituted, if the habit of justice is inculcated it touches 
a part of the being. There is an emotional response. 
We know but little of a human being, but we can safely 
say that there are depths in it, beyond the feelings of 
momentary resentment and the stimulus of pleasurable 
or painful sensation, to which justice is natural. 

How little adequate is our physical knowledge of a 
human being as a bodily frame to explain the fact of 
human life. Now and again we see one of these isolated 
beings bound up in another, as if there was an undis- 
covered physical bond between them. And in all there 
is this sense of justice — a kind of indwelling verdict of 
the universal mind, if we may use such an expression, 
in virtue of which a man feels not as a single individual 
but as all men. 

With respect to justice, it is not only necessary to 
take the view of one other person than oneself, but that 
of many. There may be justice which is very good 
justice from the point of view of a party, but very bad 
justice from the point of view of a nation. And if we 
suppose an agency outside the human race, gifted with 
intelligence, and affecting the race, in the way for instance 
of causing storms or disturbances of the ground, in order 
to judge it with justice we should have to take a stand- 
point outside the race of'men altogether. We could not 
say that this agency was bad. We should have to 
judge it with reference to its effect on other sentient 
beings. 

There are some words which are often used in contrast 



2 6 New Era of Thought. 

with each other — egoism and altruism ; and each seems 
to me unmeaning except as terms in a contrast. 

Let us take an instance. A boy has a bag of cakes, 
and is going to enjoy them by himself. His -parent 
stops him, and makes him set up some stumps and 
begin to learn to play cricket with another boy. The 
enjoyment of the cakes is lost — he has given that up ; 
but after a little while he has a pleasure which is greater 
than that of cakes in solitude. He enters into the life 
of the game. He has given up, or been forced to give 
up, the pleasure he knew, and he has found a greater 
one. What he thought about himself before was that 
he liked cakes, now what he thinks about himself is 
that he likes cricket. Which of these is the true thought 
about himself.? Neither, probably, but at any rate it 
is more near the truth to say that he likes the cricket. 
If now we use the word self to mean that which a 
person knows of himself, and it is difficult to see what 
other meaning it can have, his self as he knew it at first 
was thwarted, was given up, and through that he dis- 
covered his true self. And again with the cricket ; he 
will make the sacrifice of giving that up, voluntarily or 
involuntarily, and will find a truer self still. 

In general there is not much difficulty in making a 
boy find out that he likes cricket ; and it is quite pos- 
sible for him to eat his cakes first and learn to play 
cricket afterwards — the cricket will not come to him as 
a thwarting in any sense of what he likes better. But 
this ease in entering in to the pursuit only shows that 
the boy's nature is already developed to the level of 
enjoying the game. The distinct moral advance would 
come in such a case when something which at first was 
hard to him to do was presented to him — and the hard- 
ness, the unpleasantness is of a double kind, the giving 
up of a pursuit or indulgence to which he is accustomed, 



Theory and Practice. 27 

and the exertion of forming the habits demanded by 
the new pursuit. 

Now it is unimportant whether the renunciation is 
forced or willingly taken. But as a general rule it may 
be laid down, that by giving up his own desires as he 
feels them at the moment, to the needs and advantage 
of those around him, or to the objects which he finds 
before him demanding accomplishment, a human being 
passes to the discovery of his true self on and on. The 
process is limited by the responsibilities which a man 
finds come upon him. 

The method of moral advance is to acquire a practical 
knowledge ; he must first see what the advantage of 
some one other than himself would be, and then he 
must act in accordance with that view of things. Then 
having acted and formed a habit, he discovers a response 
in himself. He finds that he really cares, and that his 
former limited life was not really himself. His body and 
the needs of his body, so far as he can observe them, 
externally are the same as before ; but he has obtained 
an inner and unintellectual, but none the less real, 
apprehension of what he is. 

Thus altruism, or the sacrifice of egoism to others, is 
followed by a truer egoism, or assertion of self, and 
this process flashed across by the transcendent lights 
of religion, wherein, as in the sense of justice and duty, 
untold depths in the nature of man are revealed entirely 
unexpressed by the intellectual apprehension which we 
have of him as an animal frame of a very high degree 
of development, is the normal one by which from child- 
hood a human being develops into the full responsi- 
bilities of a man. 

Now both in science and in conduct there are self 
elements. In science, getting rid of the self elements 
means a truer apprehension of the facts about one ; in 



28 New Era of Thought. 

conduct, getting rid of the self elements means obtain- 
ing a truer knowledge of what we are — in the way of 
feeling more strongly and deeply and being bound and 
linked in a larger scale. 

Thus without pretending to any scientific accuracy 
in the use of terms, we can assign a certain amount of 
meaning to the expression — getting rid of self elements. 
And all that we can do is to take the rough idea of 
this process, and then taking our special subject matter, 
apply it. In affairs of life experiments lead to disaster. 
But happily science is provided wherein the desire to 
put theories into practice can be safely satisfied — and 
good results sometimes follow. Were it not for this the 
human race might before now have been utopiad from 
off the face of the earth. 

In experiment, manipulation is everything ; we must 
be certain of all our conditions, otherwise we fail as- 
suredly and have not even the satisfaction of knowing 
that our failure is due to the wrongness of our con- 
jectures. 

And for our purposes we use a subject matter so 
simple that the manipulation is easy. 



CHAPTER V. 

KNOWLEDGE : SELF-ELEMENTS. 

I MUST now go with somewhat of detail into the special 
subject in which these general truths will be exhibited. 
Everything I have to say would be conceived much 
more clearly by a very little practical manipulation. 

But here I want to put the subject in as general a 
light as possible, so that there may be no hindrance to 
the judgment of the reader. 

And when I use the word " know," I assume some- 
thing else than the possession of a rule, by which it can 
be said how facts are. By knowing I mean that the 
facts of a subject all He in the mind ready to come out 
vividly into consciousness when the attention is directed 
on them. Michael Angelo knew the human frame, he 
could tell every little fact about it ; if he chose to call 
up the image, he would see mentally how each muscle 
and fold of the skin lay with regard to the surrounding 
parts. We want to obtain a knowledge as good as 
Michael Angelo's. There is a great difference between 
Michael Angelo and us ; but let that difference be ex- 
pressed, not in our way of knowing, but in the difference 
between the things he knew and the things we know. 
We take a very simple structure and know it as abso- 
lutely as he knew the complicated structure of the 
human body.. 

And let us take a block of cubes ; any number will do, 

but for convenience sake let us take a set of twenty-seven 

29 



30 New Era of Thought. 

cubes put together so as to form a large cube of twenty- 
seven parts. And let each of these cubes be marked 
so as to be recognized, and let each have a name so that 
it can be referred to. And let us suppose that -we have 
learnt this block of cubes so that each one is known — 
that is to say, its position in the block is known and its 
relation to the other blocks. 

Now having obtained this knowledge of the block as 
it stands in front of us, let us ask ourselves if there is 
any self element present in our knowledge of it. 

And there is obviously this self element present. We 
have learnt the cubes as they stand in accordance with 
our own convenience in putting them up. We put the ■ 
lowest ones first, and the others on the top of them, 
and we distinctly conceive the lower ones as supporting 
the upper ones. Now this fact of support has nothing 
to do with the block of cubes itself, it depends on the 
conditions under which we come to apprehend the block 
of cubes, it depends on our position on the surface of 
the earth, whereby gravity is an all important factor in 
our experience. In fact our sight has got so accustomed 
to take gravity into consideration in its view of things, 
that when we look at a landscape or object with our 
head upside down we do not see it inverted, but we 
superinduce on the direct sense impressions our know- 
ledge of the action of gravity, and obtain a view differing 
very little from what we see when in an upright posi- 
tion. 

It will be found that every fact about the cubes has 
involved in it a reference to up and down. It is by 
being above or below that we chiefly remember where 
the cubes are. But above and below is a relation which 
depends simply on gravity. If it were not for gravity 
above and below would be interchangeable terms, in- 
stead of expressing a difference of marked importance 



Knowledge : Self-elements. 3 1 

to us under our conditions of existence. Now we put 
" being above " or " being below " into the cubes them- 
selves and feel it a quality in them — it defines their 
position. But this above or below really comes from 
the conditions in which we are. It is a self element, and 
as such, to obtain a true knowledge of the cubes we 
must get rid of it. 

And now, for the sake of a process which will be ex- 
plained afterwards, let us suppose that we cannot move 
the block of cubes which we have put up. Let us keep 
it fixed. 

In order to learn how it is independent of gravity the 
best way would be to go to a place where gravity has 
virtually ceased to act ; at the centre of the earth, for 
instance, or in a freely falling shell. 

But this is impossible, so we must choose another way. 
Let us, then, since we cannot get rid of gravity, see 
what we have done already. We have learnt the cubes, 
and however they are learnt, it will be found that there 
is a certain set of them round which the others are 
mentally grouped, as being on the right or left, above 
or below. Now to get our knowledge as perfect as we 
can before getting rid of the self element up and down, 
we have to take as central cubes in our mind different 
sets again and again, until there are none which are 
primary to us. 

Then there remains only the distinction of some being 
above others. Now this can only be made to sink out 
of the primary place in our thoughts by reversing the 
relation. If we turned the block upside down, and 
learnt it in this new position, then we should learn the 
position of the cubes with regard to each other with 
that element in them, which comes from the action of 
gravity, reversed. And the true nature of the arrange- 
ment to which we added something in virtue of our 



32 New Era of Thought. 

sensation of up and down, would become purer and more 
isolated in our minds. 

We have, however, supposed that the cubes are fixed. 
Then, in order to learn them, we must put up another 
block showing what they would be like in the supposed 
new position. We then take a set of cubes, models of 
the original cubes, and by consideration we can put 
them in such positions as to be an exact model of what 
the block of cubes would be if turned upside down. 

And here is the whole point on which the process 
depends. We can tell where each cube would come, 
but we do not know the block in this new position. I 
draw a distinction between the two acts, " to tell where 
it would be," and to " know." Telling where it would 
be is the preparation for knowing. The power of as- 
signing the positions may be called the theory of the 
block. The actual knowledge is got by carrying out 
the theory practically, by putting up the blocks and 
becoming able to realize without effort where each 
one is. 

It is not enough to put up the model blocks in the 
reverse position. It is found that this up and down 
is a very obstinate element indeed, and a good deal 
of work is requisite to get rid of it completely. But 
when it is got rid of in one set of cubes, the faculty 
is formed of appreciating shape independently of the 
particular parts which are above or below on first ex- 
amination. We discover in our own minds the faculty 
of appreciating the facts of position independent of 
gravity and its influence on us. I have found a very 
great difference in different minds in this respect. To 
some it is easy, to some it is hard. 

And to use our old instance, the discovery of this 
capacity is like the discovery of a love of justice in the 
being who has forced himself to act justly. It is a 



Knowledge: Self-elements. 33 

capacity for being able to take a view independent of 
the conditions under which he is placed, and to feel in 
accordance with that view. There is, so far as I know, 
no means of arriving immediately at this impartial ap- 
preciation of shape. It can only be done by, as it were, 
extending our own body so as to include certain cubes, 
and appreciating then the relation of the other cubes to 
those. And after this, by identifying ourselves with 
other cubes, and in turn appreciating the relation of the 
other cubes to these. And the practical putting up of 
the cubes is the way in which this power is gained. It 
springs up with a repetition of the mechanical acts. Thus 
there are three processes, ist. An apprehension of what 
the position of the cubes would be. 2nd, An actual put- 
ting of them up in accordance with that apprehension. 
5rd, The springing up in the mind of a direct feeling of 
what the block is, independent of any particular pre- 
sentation. 

Thus the self element of up and down can be got rid 
of out of a block of cubes. 

And when even a little block is known like this, the 
mind has gained a great deal. 

Yet in the apprehension and knowledge of the block 
of cubes with the up and down relation in them, there 
is more than in the absolute apprehension of them. For 
there is the apprehension of their position and also of 
the effect of gravity on them in their position. 

Imagine ourselves to be translated -suddenly to 
another part of the universe, and to find there intelli- 
gent beings, and to hold conversation with them. If 
we told them that we came from a world, and were to 
describe the sun to them, saying that it was a bright, 
hot body which moved round us, they would reply : 
You have told us something about the sun, but you have 
also told us something about yourselves. 

D 



34 A New Era of TJionght. 

Thus in the apprehension of thfe sun as a body moving 
round us there is more than in the apprehension of it as 
not moving round, for we really in this case apprehend 
two things — the sun .and our own conditions. But for 
the purpose of further knowledge it is most importaiit 
that the more abstract knowledge should be acquired. 
The self element introduced by the motion of the earth 
must be got rid of before the true relations of the solar 
system can be made out. 

And in our block of cubes, it will be found that feel- 
ings about arrangement, and knowledge of space, which 
are quite unattainable with our ordinary view of posi- 
tion, become simple and clear when this discipline has 
been gone through. 

And there can be no possible mental harm in going 
through this bit of training, for all. that it comes to is 
looking at a real thing as it actually is — turning it 
round and over and learning it from every point of 
view. 



CHAPTER VI. 

FUNCTION OF MIND. SPACE AGAINST METAPHYSICS. 
SELF-LIMITATION AND ITS TEST. A PLANE WORLD. 

We now pass on to the question : Are there any other 
self elements present in our knowledge of the block of 
cubes ? 

When we have learnt to free it from up and down, is 
there anything else to be got rid of .'' 

It seems as if, when the cubes were thus learnt, we had 
got as abstract and impersonal a bit of knowledge as 
possible. 

But, in reality, in the relations of the cubes as we thus 
apprehend them there is present a self element to which 
the up and down is a mere trifle. If we think we have 
got absolute knowledge we are indeed walking on a 
thin crust in unconsciousness of the depths below. 

We are so certain of that which we are habituated to, 
we are so sure that the world is made up of the me- 
chanical forces and principles which we familiarly deal 
with, that it is more of a shock than a welcome surprise 
to us to find how mistaken we were. 

And after all, do we suppose that the facts of distance 
and size and shape are the ultimate facts of the world — 
is it in truth made up like a machine out of mechanical 
parts ? If so, where is there room for that other which 
we know — more certainly, because inwardly — that reve- 
rence and love which make life worth having ? No ; 
these mechanical relations are our means of knowin 



36 A New Era of Thought. 

about the world ; they are not reality itself, and their 
primary place in our imaginations is due to the famili- 
arity which we have with them, and to the peculiar limi- 
tations under which we are. 

But I do not for a moment wish to go in thought be- 
yond physical nature — I do not suppose that in thought 
we can. To the mind it is only the body that appears, 
and all that I hope to do is to show material relations, 
mechanism, arrangements. 

But much depends on what kind of material relations 
we perceive outside us. A human being, an animal and 
a machine are to the mind all merely portions of matter 
arranged in certain ways. But the mind can give an 
exhaustive account of the machine, account fairly well 
for the animal, while the human being it only defines 
externally, leaving the real knowledge to be supplied by 
other faculties. 

But we must not under-estimate the work of the mind, 
for it is only by the observation of and thought about the 
bodies with which we come into contact that we know 
human beings. It is the faculty of thought that puts us 
in a position to recognize a soul. 

And so, too, about the universe — it is only by correct 
thought about it that we can perceive its true moral 
nature. 

And it will be found that the deadness which we 
ascribe to the external world is not really there, but is 
put in by us because of our own limitations. It is really 
the self elements in our knowledge which make us talk 
of mechanical necessity, dead matter. When our limi- 
tations fall, we behold the spirit of the world like we be- 
hold the spirit of a friend — something which is discerned 
in and through the material presentation of a body to 
us. 

Our thought means are sufficient at present to show 



space against Metaphysics. 37 

us human souls ; but all except human beings is, as far 
as science is concerned, inanimate. One self element 
must be got rid of from our perception, and this will be 
changed. 

The one thing necessary is, that in matters of thinking 
we will not admit anything that is not perfectly clear, 
palpable and evident. On the mind the only conceiv- 
able demand is to seek for facts. The rock on which so 
many systems of philosophy have come to grief is the 
attempt to put moral principles into nature. Our only 
duty is to accept what we find. Man is no more the 
centre of the moral world than he is of the physical 
world. Then relegate the intellect to its right position 
of dealing with facts of arrangement — it can appreciate 
structure — and let it simply look on the world and report 
on it. We have to choose between metaphysics and 
space thought. In metaphysics we find lofty ideals — 
principles enthroned high in our souls, but which reduce 
the world to a phantom, and ourselves to the lofty spec- 
tators of an arid solitude. On the other hand, if we 
follow Kant's advice, we use our means and find realities 
linked together, and in the physical interplay of forces 
and connexion of structure we behold the relations 
between spirits — those dwelling in man and those above 
him. 

It is difficult to explain this next self element that has 
to be removed from the block of, cubes ; it requires a 
little careful preparation, in fact our language hardly 
affords us the means. But it is possible to approach in- 
directly, and to detect the self-element by means of an 
analogy. 

If we suspect there be some condition affecting our- 
selves which make us perceive things not as they are, 
but falsely, then it is possible to test the matter by mak- 
ing the supposition of other beings subject to certain 



38 A New Era of Thought. 

conditions, and then examining what the effect on their 
experience would be of these conditions. 

Thus if we make up the appearances which would 
present themselves to a being subject to a limitation or 
condition, we shall find that this limitation or condition, 
when unrecognized by him, presents itself as a general 
law of his outward world, or as properties and qualities 
of the objects external to him. He will, moreover, find 
certain operations possible, others impossible, and the 
boundary line, between the possible and impossible will 
depend quite as much on the conditions under which he 
is as on the nature of the operations. 

And if we find that in our experience of the outward 
world there are analogous properties and qualities of 
matter, analogous possibilities and impossibilities, then 
it will show to us that we in our turn are under analo- 
gous limitations, and that what we perceive as the ex- 
ternal world is both -the external world and our own 
conditions. And the task before us will be to separate 
the two. Now the problem we take up here is this — to 
separate the self elements from the true fact. To separ- 
ate them not merely as an outward theory and intelligent 
apprehension, but to separate them in the consciousness 
itself, so that our power of perception is raised to a 
higher level. We find out that we are under limitations. 
Our next step is to so familiarize ourselves with the real 
aspect of things, that we perceive like beings not under 
our limitations. Or more truly, we find that inward 
soul which itself not subject to these limitations, is 
awakened to its own natural action, when the verdicts 
conveyed to it through the senses are purged of the self 
elements introduced by the senses. 

Everything depends on this — Is there a native and 
spontaneous power of apprehension, which springs into 
activity when we take the trouble to present to it a view 



Self-limitatioii and its Test. 39 

from which the self elements are eliminated ? About 
this every one must judge for himself. But the pro- 
cess whereby this inner vision is called on is a de- 
finite one. 

And just as a human being placed in natural human 
relationships finds in himself a spontaneous motive 
towards the fulfilment of them, discovers in himself a 
being whose motives transcend the limits of bodily self- 
regard, so we should expect to find in our minds a power 
which is ready to apprehend a more absolute order of 
fact than that which comes through the senses. 

I do not mean a theoretical power. A theory is al- 
ways about it, and about it only. I mean an inner view, 
a vision whereby the seeing mind as it were identifies 
itself with the thing seen. Not the tree of knowledge, 
but of the inner and vital sap which builds up the tree 
of knowledge. 

And if this point is settled, it will be of some use in 
answering the question : What are we .-" Are we then 
bodies only .'' This question has been answered in the 
negative by our instincts. Why should we despair of a 
rational answer ? Let us adopt our space thought and 
develop it. 

The supposition which we must make is the following. 
Let us imagine a smooth surface — like the surface of a 
table ; but let the solid body at which we are looking be 
very thin, so that our surface is more like the surface of 
a thin sheet of metal than the top of a table. 

And let us imagine small particles, like particles of 
dust, to lie on this surface, and to be attracted down- 
wards so that they keep on the surface. But let us sup- 
pose them to move freely over the surface. Let them 
never in their movements rise one over the other ; let 
them all singly and .collectively be close to the surface. 
And let us suppose all sorts of attractions and repulsions 



40 A New Era of Thought. 

between . these atoms, and let them have all kinds of 
movements like the atoms of our matter have. 

Then there may be conceived a whole world, and 
various kinds of beings as formed out of this matter. 
The peculiarity about this world and these beings would 
be, that neither the inanimate nor the animate members 
of it would move away from the surface. Their move- 
ments would all lie in one plane, a plane parallel to and 
very near the surface on which they are. 

And if we suppose a vast mass to be formed out of 
these atoms, and to lie like a great round disk on the 
surface, compact and cohering closely together, then this 
great disk would afford a support for the smaller shapes, 
which we may suppose to be animate beings. The 
smaller shapes would be attracted to the great disk, but 
would be arrested at its rim. They would tend to the 
centre of the disk, but be unable to get nearer to the 
centre than its rim. 

Thus, as we are attracted to the centre of the earth, 
but walk on its surface, the beings on this disk would be 
attracted to its centre, but walk on its rim. The force 
of attraction which they would feel would be the attrac- 
tion of the disk. The other force of attraction, acting 
perpendicularly to the plane which keeps them and all the 
matter of their world to the surface, they would know 
nothing about. For they cannot move either towards this 
force or away from it ; and the surface is quite smooth, 
so that they feel no friction in their movement over it. 

Now let us realize clearly one of these beings as he 
proceeds along the rim of his world. Let us imagine 
him in the form of an outline of a human being, with no 
thickness except that of the atoms of his world. As to 
the mode in which he walks, we must imagine that he 
proceeds by springs or hops, because there would be no 
room for his limbs to pass each other. 



Self -limitation and its Test. 41 

Imagine a large disk on the table before you, and a 
being, such as the one described, proceeding round it. 
Let there be small movable particles surrounding him, 
which move out of his way as he goes along, and let 
these serve him for respiration ; let them constitute an 
atmosphere. 

Forwards and backwards would be to such a being 
direction along the rim — the direction in which he was 
proceeding and its reverse. 

Then up and down would evidently be the direction 
away from the disk's centre and towards it. Thus back- 
wards and forwards, up and down, would both lie in the 
plane in which he was. 

And he would have no other liberty of movement 
except these. Thus the words right and left would have 
no meaning to him. All the directions in which he 
could move, or could conceive movement possible, would 
be exhausted when he had thought of the directions 
along the rim and at right angles to it, both 'in the plane. 

What he would call solid bodies, would be groups of 
the atoms of his world cohering together. Such a mass 
of atoms would, we know, have a slight thickness ; 
namely, the thickness of a single atom. But of this he 
would know nothing. He would say, " A solid body 
has two dimensions — height (by how much it goes away 
from the rim) and thickness (by how much it lies along 
the rim)." Thus a solid would be a two-dimensional 
body, and a solid would be bounded by lines. Lines 
would be all that he could see of a solid body. 

Thus one of the results of the limitations under which 
he exists would be, that he would say, " There are only 
two dimensions in real things." 

In order for his world to be permanent, we must 
suppose the surface on which he is to be very compact, 
compared to the particles of his matter ; to- be very 



42 



A New Era of Thought. 



rigid ; and, if he is not to observe it by the friction of 
matter moving on it, to be very smooth. And if it is 
very compact with regard to his matter, the vibrations of 
the surface must have the effect of disturbing the portions 
of his matter, and of separating compound bodies up 
into simpler ones. 

Another consequence of the limitation under which 
this being lies, would be the following : — If we cut out 




from the corners of a piece of paper two triangles, ABC 
and A' B' C, and suppose them to be reduced to such 
a thinness that they are capable of being put on to the 
imaginary surface, and of being observed by the flat, 
being like other bodies known to him ; he will, after 
studying the bounding lines, which are all that he can see 
or touch, come to the conclusion that they are equal and 
similar in every respect ; and he can conceive the one 
occupying the same space as the other occupies, without 
its being altered in any way. 

If, however, instead of putting down these triangles 
into the surface on which the supposed being lives, as 
shown in Fig. i, we first of all turn one of them over. 



A Plane World. 43 

and then put them down, then the plane-being has pre- 
sented to him two triangles, as shown in Fig. 2. 

And if he studies these, he finds that they are equal 
in size and similar in every respect. But he cannot 
make the one occupy the same space as the other one ; 
this will become evident if the triangles be moved about 
on the surface of a table. One will not lie on the same 
portion of the table that the other has marked out by 
lying on it. 

Hence the plane-being by no means could make the 
one triangle in this case coincide with the space occupied 
by the other, nor would he be able to conceive the one 
as coincident with the other. 

The reason of this impossibility is, not that the one 
cannot be made to coincide, but that before having been 
put down on his plane it has been turned round. It 
has been turned, using a direction of motion which the 
plane-being has never had any experience of, and which 
therefore he cannot use in his mental work any more 
than in his practical endeavours. 

Thus, owing to his limitations, there is a certain line 
of possibility which he cannot overstep. But this line 
does not correspond to what is actually possible and 
impossible. It corresponds to a certain condition affect- 
ing him, not affecting the triangle. His saying that it 
is impossible to make the two triangles coincide, is an 
assertion, not about the triangles, but about himself 

Now, to return to our own world, no doubt there are 
many assertions which we make about the external 
world which are really assertions about ourselves. And 
we have a set of statements which are precisely similar 
to those which the plane-being would make about his 
surroundings. 

Thus, he would say, there are only two independent 
directions ; we say there are only three. 



44 A. New Era of Thought. 

He would say that solids are bounded by lines ; we 
say that solids are bounded by planes. 

Moreover, there are figures about which we assert 
exactly the same kind of impossibility as his plane-being 
did about the triangles in Fig. 2. 

We know certain shapes which are equal the one to 
the other, which are exactly similar, and yet which we 
cannot make fit into the same portion of space, either 
practically or by imagination. 

If we look at our two hands we see this clearly, 
though the two hands are a complicated case of a very 
common fact of shape. Now, there is one way in 
which the right hand and the left hand may practically 
be brought into likeness. If we take the right-hand 
glove and the left-hand glove, they will not fit any more 
than the right hand will coincide with the left hand. 
But if we turn one glove inside out, then it will fit. Now, 
to suppose the same thing done with the solid hand as 
is done with the glove when it is turned inside out, we 
must suppose it, so to speak, pulled through itself. If 
the hand were inside the glove all the time the glove 
was being turned inside out, then, if such an operation 
were possible, the right hand would be turned into an 
exact model of the left hand. Such an operation is 
impossible. But curiously enough there is a precisely 
similar operation which, if it were possible, would, in a 
plane, turn the one triangle in Fig. 2 into the exact 
copy of the other. 

Look at the triangle in Fig. 2, ABC, and imagine 
the point A to move into the interior of the triangle and 
to pass through it, carrying after it the parts of the lines 
A B and A C to which it is attached, we should have 
finally a triangle ABC, which was quite like the other 
of the two triangles A' B' C in Fig. 2. 

Thus we know the operation which produces the 



A Plane World. 



45 



result of the " pulling through " is not an impossible one 
when the plane-being is concerned. Then may it not be 
that there is a way in which the results of the impossible 
operation of pulling a hand through could be performed ? 
The question is an open one. Our feeling of it being 
impossible to produce this result in any way, may be 
because it really is impossible, or it may be a useful bit 
of information about ourselves. 

Now at this point my special work comes in. If there 
be really a four-dimensional world, and we are limited 









to a space or three-dimensional view,^ then either we are 
absolutely three-dimensional with no experience at all 
or capacity of apprehending four-dimensional facts, or 
we may be, as far as our outward experience goes, so 
limited ; but we may really be four-dimensional beings 
whose consciousness is by certain undetermined con- 
ditions limited to a section of the real space. 

Thus we may really be like the plane-beings mentioned 
above, or we may be in such a condition that our percep- 
tions, not ourselves, are so limited. The question is one 
which calls for experiment. 

We know that if we take an animal, such as a dog 



46 A New Era of TJiought. 

or cat, we can by, careful training, and by using rewards 
and punishment, make them act in a certain way, in 
certain defined cases, in accordance with justice ; we 
can produce the mechanical action. But the feeling 
of justice will not be aroused ; it will be but a mere 
outward conformity. But a human being, if so trained, 
and seeing others so acting, gets a feeling of justice. 

Now, if we are really four-dimensional, by going 
through those acts which correspond to a four-dimen- 
sional experience (so far as we can), we shall obtain an 
apprehension of four-dimensional existence — not with 
the outward eye, but essentially with the mind. 

And after a number of years of experiment which were 
entirely nugatory, I can now lay it down as a verifiable 
fact, that by taking the proper steps we can feel four- 
dimensional existence, that the human being somehow, 
and in some way, is not simply a three-dimensional 
being — in what way it is the province of science to 
discover. All that I shall do here is, to put forward 
certain suppositions which, in an arbitrary and forced 
manner, give an outline of the relation of our body to 
four-dimensional existence, and show how in our minds 
we have faculties by which we recognise it. 



CHAPTER VII. 

SELF ELEMENTS IN OUR CONSCIOUSNESS. 

It is often taken for granted that our consciousness of 
ourselves and of our own feelings has a sort of direct 
and absolute value. 

It is supposed to afford a testimony which does not 
require to be sifted like our consciousness of external 
events. But in reality it needs far more criticism to be 
applied to it than any other mode of apprehension. 

To a certain degree we can sift our experience of 
the external world, and divide it into two portions. 
We can determine the self elements and the realities. 
But with regard to our own nature and emotions, the 
discovery which makes a science possible has yet to be 
made. 

There are certain indications, however, springing from 
our observation of our own bodies, which have a certain 
degree of interest. 

It is found that the processes of thought and feeling 
are connected with the brain. If the brain is disturbed, 
thoughts, sights, and sounds come into the conscious- 
ness which have no objective cause in the external 
world. Hence we may conclusively say that the human 
being, whatever he is, is in contact with the brain, and 
through the brain with the body, and through the body 
with the external world. 

It is the structures and movements in the brain which 



48 A New Era of Thotight. 

the human being perceives. It is by a structure in the 
brain that he apprehends nature, not immediately. 
The most beautiful sights and sounds have no effect 
on a human being unless there is the faculty in the 
brain of taking them in and handing them on to the 
consciousness. 

Hence, clearly, it is the movements and structure of 
the minute portions of matter forming the brain which 
the consciousness perceives. And it is only by models 
and representations made in the stuff of the brain that 
the mind knows external changes. 

Now, our brains are well furnished with models and 
representations of the facts and events of the external 
world. 

But a most important fact still requires its due weight 
to be laid upon it. 

These models and representations are made on a very 
minute scale — the particles of brain matter which form 
images and representations are beyond the power of the 
microscope in their minuteness. Hence the conscious- 
ness primarily apprehends the movements of matter of 
a degree of smallness which is beyond the power of 
observation in any other way. 

Hence we have a means of observing the movements 
of the minute portions of matter. Let us call those 
portions of the brain matter which are directly instru- 
mental in making representations of the external world 
— let us call them brain molecules. 

Now, these brain molecules are very minute portions 
of matter indeed ; generally they are made to go 
through movements and form structures in such a way 
as to represent the movements and structures of the 
external world of masses around us. 

But it does not follow that the structures and move- 
ments which they perform of their own nature are 



Self Elements in our Consciousness. 49 

identical with the movements of the portions of matter 
which we see around us in the world of matter. 

It may be that these brain molecules have the power 
of four-dimensional movement, and that they can go 
through four-dimensional movements and form four- 
dimensional structures. 

If so, there is a practical way of learning the move- 
ments of the very small particles of matter — by observ- 
ing, not what we can see, but what we can think. 

For, suppose these small molecules of the brain were 
to build up structures and go through movements not 
in accordance with the rule of representing what goes 
on in the external world, but in accordance with their 
own activity, then they might go through four-dimen- 
sional movements and form four-dimensional structures. 

And these movements and structures would be ap- 
prehended by the consciousness along with the other 
movements and structures, and would seem as real as 
the others — but would have no correspondence in the 
external world. 

They would be thoughts and imaginations, not ob- 
servations of external facts. 

Now, this field of investigation is one which requires 
to be worked at. 

At present it is only those structures and movements 
of the brain molecules which correspond to the realities 
of our three-dimensional space which are in general 
worked at consistently. But in the practical part of 
this book it will be found that by proper stimulus the 
brain molecules will arrange themselves in structures 
representing a four-dimensional existence. It only 
requires a certain amount of care to build up mental 
models of higher space existences. In fact, it is pro- 
bably part of the difficulty of forming three-dimensional 
brain models, that the brain molecules have to be limited 

E 



5° A New Era of Thought . 

in their own freedom of motion to the requirements of 
the limited space in which our practical daily life is 
carried on. 

Noie. — For my own part I should say that all those confusions in 
remembering which come from an image taking the place of the 
original mental model — as, for instance, the difficulty in remember- 
ing which way to turn a screw, and the numerous cases of images 
in thought transference — may be due to a toppling over in the 
brain, four-dimensionalwise, of the structures formed — which 
structures would be absolutely safe from being turned into image 
structures if the brain molecules moved only three-dimensional- 
wise. 

It is remarkable how in science " explaining " means 
the reference of the movements and tendencies to 
movement of the masses about us to the movements 
and tendencies to movement of the minute portions of 
matter. 

Thus, the behaviour of gaseous bodies — the pressure 
which they exert, the laws of their cooling and inter- 
mixture are explained by tracing the movements of the 
very minute particles of which they are composed. 



CHAPTER VIII. 

RELATION OF LOWER TO HIGHER SPACE. THEORY 
OF THE ^THER. 

At this point of our inquiries the best plan is to turn 
to the practical work, and try if the faculty of thinking 
in higher space can be awakened in the mind. 

The general outline of the nnethod is the same as that 
which has been described for getting rid of the liriiita- 
tion of up and down from a block of cubes. We sup- 
posed that the block was fixed ; and to get the sense of 
what it would be when gravity acted in a different way 
with regard to it, we made a model of it as it would be 
under the new circumstances. We thought out the 
relations which would exist ; and by practising this new 
arrangement we gradually formed the direct appre- 
hension. 

And so with higher-space arrangements. We cannot 
put them up actually, but we can say how they would 
look and be to the touch from various sides. And we 
can put up the actual appearances of them, not alto- 
gether, but as models succeeding one another ; and by 
contemplation and active arrangement of these different 
views we call upon our inward power to manifest itself. 

In preparing our general plan of work, it is necessary 
to make definite assumptions with regard to our world, 
our universe, or we may call it our space, in relation to 
the wider universe of four-dimensional space. 

What our relation to it may be, is altogether un- 
determined. The real relationship will require a great 



52 A New Era of Thovght. 

deal of study to apprehend, and when apprehended will 
seem as natural to us as the position of the earth among 
the other planets does to us now. 

But we have not got to wait for this exploration in 
order to commence our work of higher-space thought, 
for we know definitely that whatever our real physical 
relationship to this wider universe may be, we are practi- 
cally in exactly the same relationship to it as the 
creature we have supposed living on the surface of a 
smooth sheet is to the world of threefold space. 

And this relationship of a surface to a solid or of a 
solid, as we conjecture, to a higher solid, is one which 
we often find in nature. A surface is nothing more nor 
less than the relation between two things. Two bodies 
touch each other. The surface is the relationship of one 
to the other. 

Again, we see the surface of water. 

Thus our solid existence may be the contact of two 
four-dimensional existences with each other; and just as 
sensation of touch is limited to the surface of the body, 
so sensation on a larger scale may be limited to this 
solid surface. 

And it is a fact worthy of notice, that in the surface 
of a fluid different laws obtain from those which hold 
throughout the mass. There are a whole series of facts 
which are grouped together under the name of surface 
tensions, which are of great importance in physics, and 
by which the behaviour of the surfaces of liquids is 
governed. 

And it may well be that the laws of our universe are 
the surface tensions of a higher universe. 

But these expressions, it is evident, afford us no practi- 
cal basis for investigation. We must assume something 
more definite, and because more definite (in the absence 
of details drawn from experience), more arbitrary. 



Relation of Lower to Higher Space. 53 

And we will assume that the conditions under which 
we human beings are, exactly resemble those under 
which the plane-beings are placed, which have been 
described. 

This forms the basis of our work ; and the practical 
part of it consists in doing, with regard to higher 
space, that which a plane-being would do with regard 
to our space in order to enable himself to realize what 
it was. 

If we imagine one of these limited creatures whose 
life is cramped and confined studying the facts of space 
existence, we find that he can do it in two ways. He 
can assume another direction in addition to those which 
he knows ; and he can, by means of abstract reasoning, 
say what would take place in an ampler kind of space 
than his own. All this would be formal work. The 
conclusions would be abstract possibilities. 

The other mode of study is this. He can take some 
of these facts of his higher space and he can ponder 
■over them in his mind, and can make up in his plane 
world those different appearances which one and the 
same solid body would present to him, and then he may 
try to realize inwardly what his higher existence is. 

Now, it is evident that if the creature is absolutely 
confined to a two-dimensional existence, then anything 
-more than such existence will always be a mere abstract 
and formal consideration to him. 

But if this higher-space thought becomes real to him, 
if he finds in his mind a possibility of rising to it, then 
indeed he knows that somehow he is not limited to his 
apparent world. Everything he sees and comes into 
contact with may be two-dimensional ; but essentially, 
somehow, himself he is not two-dimensional merely. 

And a precisely similar piece of work is before us. 
Assuming as we must that our outer experience is 



54 A New Era of Thought. 

limited to three-dimensional space, we shall make up 
the appearances which the very simplest higher bodies 
would present to us, and we shall gradually arrive at a 
more than merely formal and abstract appreciation of 
them. We shall discover in ourselves a faculty of ap- 
prehension of higher space similar to that which we have 
of space. And thus we shall discover, each for himself, 
that, limited as his senses arc, he essentially somehow 
is not limited. 

The mode and method in which this consciousness 
will be made general, is the same in which the spirit of 
an army is formed. 

The individuals enter into the service from various 
motives, but each and all have to go through those 
movements and actions which correspond to the unity 
of a whole formed out of different members. The inner 
apprehension which lies in each man of a participation 
jn a life wider than that of his individual body, is 
awakened and responds ; and the active spirit of the 
army is formed. So with regard to higher space, this 
faculty of apprehending intuitively four-dimensional 
relationships will be taken up because of its practical 
use. Individuals will be practically employed to do it 
by society because of the larger faculty of thought 
which it gives. In fact, this higher-space thought means 
as an affair of mental training simply the power of ap- 
prehending the results arising from four independent 
causes. It means the power of dealing with a greater 
number of details. 

And when this faculty of higher-space thought has 
been formed, then the faculty of apprehending that 
higher existence in which men have part, will come 
into being. 

It is necessary to guard here against there being 
ascribed to this higher-space thought any other than 



Relation of Lower to Higher Space. 55 

an intellectual value. It has no moral value whatever. 
Its only connexion with moral or ethical considerations 
is the possibility it will afford of recognizing more of 
the facts of the universe than we do now. There is a 
gradual process going on which may be described as 
the getting rid of self elements. This process is one of 
knowledge and feeling, and either may be independent 
of the other. At present, in respect of feeling, we are 
much further on than in respect to understanding, and 
the reason is very much this : When a self element has 
been got rid of in respect of feeling, the new appre- 
hension is put into practice, and we live it into our 
organization. But when a self element has been got rid 
of intellectually, it is allowed to remain a matter of 
theory, not vitally entering into the mental structure of 
individuals. 

Thus up and down was discovered to be a self element 
more than a thousand years ago ; but, except as a matter 
of theory, we are perfect barbarians in this respect up to 
the present day. 

We have supposed a being living in a plane world, 
that is, a being of a very small thickness in a direction 
perpendicular to the surface on which he is. 

Now, if we are situated analogously with regard to 
an ampler space, there must be some element in our 
experience corresponding to each element in the plane- 
being's experience. 

And it is interesting to ask, in the case of the plane- 
being, what his opinion would be with respect to the 
surface on which he was. 

He would not recognize it as a surface with which 
he was in contact ; he would have no idea of a motion 
away from it or towards it. 

But he would discover its existence by the fact that 
movements were transmitted along it. By its vibrating 



56 A New Era of Thoiight. 

and quivering, it would impart movement to the par- 
ticles of matter lying on it. 

Hence, he would consider this surface to be a medium 
lying between bodies, and penetrating them. It would 
appear to him to have no weight, but to be a powerful 
means of transmitting vibrations. Moreover, it would 
be unlike any other substance with which he was 
acquainted, inasmuch as he could never get rid of 
it. However perfect a vacuum be rnade, there would 
be in this vacuum just as much of this unknown me- 
dium as there was before. 

Moreover, this surface would not hinder the move- 
ment of the particles of matter over it. Being smooth, 
matter would slide freely over it. And this would seem 
to him as if matter went freely through the medium. 

Then he would also notice the fact that vibrations 
of this medium would tear asunder portions of matter. 
The plane surface, being very compact, compared to 
the masses of matter on it, would, by its vibrations, 
shake them into their component parts. 

Hence he would have a series of observations which 
tended to show that this medium was unlike any or- 
dinary matter with which he was acquainted. Although 
matter passed freely through it, still by its shaking it 
could tear matter in pieces. These would be very 
difficult properties to reconcile in one and the same 
substance. Then it is weightless, and it is everywhere. 

It might well be that he would regard the 'suppo- 
sition of there being a plane surface, on which he was, 
as a preferable one to the hypothesis of this curious 
medium ; and thus he might obtain a proof of his limi- 
tations from his observations. 

Now, is there anything in our experience which 
corresponds to this medium which the plane-being gets 
to observe ? 



Theory of the ^ther, 5 7 

Do we suppose the existence of any medium through 
which matter freely moves, which yet by its vibrations 
destroys the combinations of matter — some medium 
which is present in every vacuum, however perfect, 
which penetrates all bodies, and yet can never be laid 
hold of ? 

These are precisely observations which have been 
made. 

The substance which possesses all these qualities is 
called the £Ether. And the properties of the aether are 
a perpetual object of investigation in science. 

Now, it is not the place here to go into details, as 
all we want, is a basis for work ; and however arbitrary- 
it may be, it will serve if it enables us to investigate 
the properties of higher space. 

We will suppose, then, that we are not in, but on the 
aether, only not on it in any known direction, but that 
the new direction is that which comes in. The aether 
is a smooth body, along which we slide, being distant 
from it at every point about the thickness of an atom ; 
or, if we take our mean distance, being distant from 
it by half the thickness of an atom measured in this 
new direction. 

Then, just as in space objects, a cube, for instance, 
can stand on the surface of a table, or on the surface 
over which the plane-being moves, so on the aether can 
stand a higher solid. 

All that the plane-being 'sees or touches of a cube, 
is the square on which it rests. 

So all that we could see or touch of a higher solid 
would be that part by which it stood on the aether ; 
and this part would be to us exactly like any ordinary 
solid body. The base of a cube would be to the 
plane-being like a square which is to him an ordinary 
solid. 



58 A New Era of Thought, 

Now, the two ways, in which a plane-being would 
apprehend a solid body, would be by the successive 
appearances to him of it as it passed through his plane ; 
and also by the different views of one and the same solid 
body which he got by turning the body over, so that 
different parts of its surface come into contact with his 
plane. 

And the practical work of learning to think in four- 
dimensional space, is to go through the appearances 
which one and the same higher solid has. 

Often, in the course of investigation in nature, we 
come across objects which have a certain similarity, and 
yet which are in parts entirely different. The work of the 
mind consists in forming an idea of that whole in which 
they cohere, and of which they are simple presentations. 

The work of forming an idea of a higher solid is the 
most simple and most definite of all such mental 
operations. 

If we imagine a plane world in which there are 
objects which correspond to our sun, to the planets, and, 
in fact, to all our visible universe, we must suppose a 
surface of enormous extent on which great disks slide, 
these disks being worlds of various orders of magni- 
tude. 

These disks would some of them be central, and hot, 
like our sun ; round them would circulate other disks, 
like our planets. 

And the systems of sun and planets must be con- 
ceived as moving with great velocity over the surface 
which bears them all. 

And the movements of the atoms of these worlds 
will be the course of events in such worlds. As the 
atoms weave together, and form bodies altering, be- 
coming, and ceasing, so will bodies be formed and 
disappear. 



Theory of the ^ther. 59 

And the plane which bears them all on its smooth 
surface will simply be a support to all these movements, 
and influence them in no way. 

Is to be conscious of being conscious of being hot, 
the same thing as to be conscious of being hot ? It is 
not the same. There is a standing outside, and objecti- 
vation of a state of mind which every one would say in 
the first state was very different from the simple con- 
sciousness. But the consciousness must do as much in 
the first case as in the second. Hence the feeling hot 
is very different from the consciousness of feeling hot. 

A feeling which we always have, we should not be 
conscious of — a sound always present ceases to be heard. 
Hence consciousness is a concomitant of change, that 
is, of the contact between one state and another. 

If a being living on such a plane were to investigate 
the properties, he would have to suppose the solid to 
pass through his plane in order to see the whole of 
its surface. Thus we may imagine a cube resting on 
a table to begin to penetrate through the table. If the 
cube passes through the surface, making a clean cut all 
round it, so that the plane-being can come up to it and 
investigate it, then the different parts of the cube as it 
passes through the plane will be to him squares, which 
he apprehends by the boundary lines. The cut which 
there is in his plane must be supposed not to be noticed, 
he must be able to go right up to the cube without hin- 
drance, and to touch and see that thin slice of it which 
is just above the plane. 

And so, when we study a higher solid, we must sup- 
pose that it passes through the aether, and that we only 
see that thin three-dimensional section of it which is 
just about to pass from one side to the other of the 
aether. 

When we look on a solid as a section of a higher 



6o A New Era of Thought. 

solid, we have to suppose the aether broken through, 
only we must suppose that it runs up to the edge of the 
body which is penetrating it, so that we are aware of 
no breach of continuity. 

The surface of the aether must then be supposed to 
have the properties of the surface of a fluid ; only, of 
course, it is a solid three-dimensional surface, not a two- 
dimensional surface. 



CHAPTER IX. 

ANOTHER VIEW OF THE .ETHER. MATERIAL AND 
^THERIAL BODIES. 

We have supposed in the case of a plane world that the 
surface on which the movements take place is inactive, 
except by its vibrations. It is simply a smooth support. 

For the sake of simplicity let us call this smooth 
surface " the sether " in the case of a plane world. 

The Eether then we have imagined to be simply a 
smooth, thin sheet, not possessed of any definite struc- 
ture, but excited by real disturbances of the matter on 
it into vibrations, which carry the effect of these dis- 
turbances as light and heat to other portions of matter. 
Now, it is possible to take an entirely different view of 
the aether in the case of a plane world. 

Let us imagine that, instead of the aether being a 
smooth sheet serving simply as a support, it is de- 
finitely marked and grooved. 

Let us imagine these grooves and channels to be very 
minute, but to be definite and permanent. 

Then, let us suppose that, instead of the matter which 
slides in the aether having attractions and repulsions of 
its own, that it is quite inert, and has only the properties 
of inertia. 

That is to say, taking a disk or a plane world as a 
specimen, the whole disk is sliding on the aether in 
virtue of a certain momentum which it has, and certain 
portions of its matter fit into the grooves in the aether, 
and move along those grooves. 

The size of the portions is determined by the size of 



62 A New Era of Thought. 

the grooves. And let us call those portions of matter 
which occupy the breadth of a groove, atoms. Then it 
is evident that the disk sliding along over the aether, its 
atoms will move according to the arrangement of the 
grooves over which the disk slides. If the grooves at 
any one particular place come close together, there will 
be a condensation of matter at that place when the 
disk passes over it ; and if the grooves separate, there 
will be a rarefaction of matter. 

If we imagine five particles, each slipping along in its 
own groove, if the particles are arranged in the form of 
a regular pentagon, and the grooves are parallel, then 
these five particles, moving evenly on, will maintain 
their positions with regard to one another, and a body 
would exist like a pentagon, lasting as long as the 
groves remained parallel. 

But if, after some distance had been traversed by the 
disk, and these five particles were brought into a region 
where one of the grooves tended away from the others, 
the shape of the pentagon would be destroyed, it would 
become some irregular figure. And it is easy to see 
that if the grooves separated, and other grooves came 
in amongst them, along which other portions of matter 
were sliding, that the pentagon would disappear as an 
isolated body, that its constituent matter would be 
separated, and that its particles would enter into other 
shapes as constituents of them, and not of the original 
pentagon. 

Thus, in cases of greater complication, an elaborate 
structure may be supposed to be formed, to alter, and to 
pass away ; its origin, growth, and decay being due, not 
to any independent motion of the particles constituting 
it, but to the movement of the disk whereby its portions 
of matter were brought to regions where there was a 
particular disposition of the grooves. 



Another View of the ^ther. 63 

Then the nature of the shape would really be deter- 
mined by the grooves, not by the portions of matter 
which passed over them — they would become manifest 
as giving rise to a material form when a disk passed 
over them, but they would subsist independently of the 
disk ; and if another disk were to pass over the same 
grooves, exactly the same material structures would 
spring up as came into being before. 

If we make a similar supposition about our sether 
along which our earth slides, we may conceive the 
movements of the particles of matter to be determined, 
not by attractions or repulsions exerted on one another, 
but to be set in existence by the alterations in the 
directions of the grooves of the sether along which 
they are proceeding. 

If the grooves were all parallel, the earth would pro- 
ceed without any other motion than that of its path in 
the heavens. 

But with an alteration in the direction of the grooves, 
the particles, instead of proceeding uniformly with the 
mass of the earth, would begin to move amongst each 
other. And by a sufficiently complicated arrangement 
of grooves it may be supposed that all the movements 
of the forms we see around us are due to interweaving 
and variously disposed grooves. 

Thus the movements, which any body goes through, 
would depend on the arrangement of the aethereal 
grooves along which it was passing. As long as the 
grooves remain grouped together in approximately the 
same way, it would maintain its existence as the same 
body ; but when the grooves separated, and became in- 
volved with the grooves of other objects, this body 
would cease to exist separately. 

Thus the separate existences of the earth might con- 
ceivably be due to the disposition of those parts of the 



64 A New Era of Thought. 

aether over which the earth passed. And thus any 
object would have to be separated into two parts, one 
the aethereal form, or modification which lasted, the 
other the material particles which, coming on with 
blind momentum, were directed into such movements as 
to produce the actual objects around us. 

In this way there would be two parts in any organism, 
the material part and the sethereal part. There would 
be the material body, which soon passes and becomes 
indistinguishable from any other material body, and the 
sethereal body which remains. 

Now, if we direct our attention to the material body, 
we see the phenomena of growth, decay, and death, the 
coming and the passing away of a living being, isolated 
during his existence, absolutely merged at his death into 
the common storehouse of matter. 

But if we regard the aethereal body, we find something 
different. We find an organism which is not so abso- 
lutely separated from the surrounding organisms — an 
organism which is part of the aether, and which is linked 
to other ethereal organisms by its very substance — an 
organism between which and others there exists a unity 
incapable of being broken, and a common life which is 
rather marked than revealed by the matter which passes 
over it. The aethereal body moreover remains per- 
manently when the material body has passed away. 

The correspondences between the aethereal body and 
the life of an organism such as we know, is rather to be 
found in the emotional region than in the one of out- 
ward observation. To the aethereal form, all parts of it 
are equally one ; but part of this form corresponds to 
the future of the material being, part of it to his past. 
Thus, care for the future and regard for the past would 
be the way in which the material being would exhibit 
the unity of the Ethereal body, which is both his- past, 



Material and /^thereat Bodies. 65 

his present, and his future. That is to say, suppose the 
asthereal body capable of receiving an injury, an injury 
in one part of it would correspond to an injury in a 
man's past ; an injury in another part, — that which the 
material body was traversing, — would correspond to an 
injury to the man at the present moment ; injury to the 
aethereal body at another part, would correspond to 
injury coming to the man at some future time. And 
the self-preservation of the sethereal body, supposing it 
to have such a motive, would in the last case be the 
motive of regarding his own future to the man. And 
inasmuch as the man felt the real unity of his aethereal 
body, and did not confine his attention to his material 
body, which is absolutely disunited' at every moment 
from its future and its past — inasmuch as he apprehended 
his aethereal unity, insomuch would he care for his future 
, welfare, and consider it as equal in importance to his 
present comfort. The correspondence between emotion 
and physical fact would be, that the emotion of regard 
corresponded to an undiscerned aethereal unity. And 
then also, just as the two tips of two fingers put down 
on a plane, would seem to a plane-being to be two com- 
pletely different bodies, not connected together, so one 
and the same aethereal' body might appear as two 
distinct material b9dies, and any regard between the 
two would correspond to an apprehension of their 
aethereal unity. In the supposition of"an aethereal body, 
it is not necessary to keep to the idea of the rigidity and 
permanence of the grooves defining the motion of the 
matter which, passing along, exhibits the material body. 
The aethereal body may have a life of its own, relations 
with other aethereal bodies,, and a life as full of vicissi- 
tudes as that of the material body, which in its total 
orbit expresses in the movements of matter one phase 
in the life of the aethereal body. 

F 



66 A New Era of Thotight. 

But there are certain obvious considerations which 
prevenf any serious dwelling on these speculations — they 
are only introduced here in order to show how the con- 
ception of higher space lends itself to the representation 
of certain indefinite apprehensions, — such as that of the 
essential unity of the race, — and affords a possible clue 
to correspondences between the emotional and the 
physical life. 

The whole question of our relation to the tether has 
to be settled. That which we call the aether is far more 
probably the surface of a liquid, and the phenomena we 
observe due to surface tensions. Indeed, the physical 
questions concern us here nothing at all. It is easy 
enough to make some supposition which gives us a 
standing ground to discipline our higher-space percep- 
tion ; and when that is trained, we shall turn round and 
look at the facts. 

The conception which we shall form of the universe 
will undoubtedly be as different from our present one, 
as the Copernican view differs from the more pleasant 
view of a wide immovable earth beneath a vast vault. 
Indeed, any conception of our place in the universe will 
be more agreeable than the thought of being on a 
spinning ball, kicked into space without any means of 
communication with any other inhabitants of the 
universe. 



CHAPTER X. 

HIGHER SPACE AND HIGHER BEING. PERCEPTION AND 
INSPIRATION. 

In the instinctive and sense perception of man and 
nature there is all hidden, which reflection afterwards 
brings into consciousness. 

We are conscious of somewhat higher than each 
individual man when we look at men. In some, this 
consciousness reaches an extreme pitch, and becomes 
a religious apprehension. But in none is it otherwise 
than instinctive. The apprehension is sufficiently defi- 
nite to be certain. But it is not expressible to us in 
terms of the reason. 

Now, I have shown that by using the conception of 
higher space it is easy enough to make a supposition 
which shall show all mankind as physical parts of one 
whole. Our apparent isolation as bodies from each 
other is by no means so necessary to assume as it 
would appear. But, of course, a supposition of that 
kind is of no value, except as showing a possibility. 
If we came to examine into the matter closely, we 
should find a natural relationship which accounted for 
our consciousness being limited as at present it is. 

The first thing to be done, is to organize our higher- 
space perception, and then look. We cannot tell what 
external objects will blend together into the unity of a 
higher being. But just as the riddle of the two hands 
becomes clear to us from our first inspection of higher 
space, so will there grow before our e\'es greater unities 

and greater surprises. 

67 



68 A New Era of Thought. 

We have been subject to a limitation of the most 
absurd character. Let us open our eyes and see the 
facts. 

Now, it requires some training to open the eyes. 
For many years I worked at the subject without the 
sHghtest success. All was mere formalism. But by 
adopting the simplest means, and by a more thorough 
knowledge of space, the whole flashed clear. 

Space shapes can only be symbolical of four-dimen- 
sional shapes ; and if we do not deal with space shapes 
directly, but only treat them by symbols on the plane — 
as in analytical geometry — we are trying to get a per- 
ception of higher space through symbols of symbols, 
and the task is hopeless. But a direct study of space 
leads us to the knowledge of higher space. And with 
the knowledge of higher space there come into our ken 
boundless possibilities. All those things may be real, 
whereof saints and philosophers have dreamed. 

Looking on the fact of life, it has become clear to 
the human mind, that justice, truth, purity, are to be 
sought — that they are principles which it is well to 
serve. And men have invented an abstract devotion 
to these, and all comes together in the grand but vague 
conception of Duty. 

But all these thoughts are to those which spring up 
before us as the shadow on a bank of clouds of a great 
mountain is to the mountain itself On the piled-up 
clouds falls the shadow — vast, imposing, but dark, colour- 
less. If the beholder but turns, he beholds the mountain 
itself, towering grandly with verdant pines, the snowline, 
and the awful peaks. 

So all these conceptions are the way in which now, 
with vision confined, we apprehend the great existences 
of the universe. Instead of an abstraction, what we 
have to serve is a reality, to which even our real things 



Higher Space and Higher Being. 69 

are but shadows. We are parts of a great being, in 
whose service, and with whose love, the utmost demands 
of duty are satisfied. 

How can it not be a struggle, when the claims of 
righteousness mean diminished life, — even death, — to 
the individual who strives .' And yet to a clear and 
more rational view it will be seen that in his extinction 
and loss, that which he loves, — that real being which 
is to him shadowed forth in the present existence of 
wife and child, — that being lives more truly, and in its 
life those he loves are his for ever. 

But, of course, there are mistakes in what we con- 
sider to be our duty, as in everything else ; and this is an 
additional reason for pursuing the quest of this reality. 
For by the rational observance of other material bodies 
than our own, we come to the conclusion that there 
are other beings around us like ourselves, whom we 
apprehend in virtue of two processes — the one simply 
a sense one of observation and reflection — ^the other a 
process of direct apprehension. 

Now, if we did not go through the sense process of 
observation, we might, it is true, know that there were 
other human beings around us in some subtle way — in 
some mesmeric feeling ; but we should not have that 
organized human life which, dealing with the things of 
the world, grows into such complicated forms. We 
should for ever be good-humoured babies — a sensuous, 
affectionate kind of jelly-fish. 

And just so now with reference to the high intelli- 
gences by whom we are surrounded. We feel them, 
but we do not realize them. 

To realize them, it will be necessary to develop our 
power of perception. 

The power of seeing with our bodily eye is limited to 
the three-dimensional section. 



JO A New Era of Thought. 

But I have shown that the inner eye is not thus 
limited ; that we can organize our power of seeing in 
higher space, and that we can form conceptions of 
realities in this higher space, just as we can in our ordi- 
nary space. 

And this affords the groundwork for the perception 
and study of these other beings than man. Just as some 
mechanical means are necessary for the apprehension 
of our fellows in space, so a certain amount of me- 
chanical education is necessary for the perception of 
higher beings in higher space. 

Let us turn the current of our thought right round ; 
instead of seeking after abstractions, and connecting our 
observations by ideas, let us train our sense of higher 
space and build up conceptions of greater realities, more 
absolute existences. 

It is really a waste of time to write or read more 
generalities. Here is the grammar of the knowledge of 
higher being — let us learn it, not spend time in specu- 
lating as to whither it will lead us. 

Yet one thing more. We are, with reference to the 
higher things of life, like blind and puzzled children. 
We know that we are members of one body, limbs of 
one vine ; but we cannot discern, except by instinct and 
feeling, what that body is, what the vine is. If to know 
it would take away our feeling, then it were well never 
to know it. But fuller knowledge of other human beings 
does not take away our love for them ; what reason is 
there then to suppose that a knowledge of the higher 
existences would deaden our feelings .' 

And then, again, we each of us have a feeling that we 
ourselves have a right to exist. We demand our own 
perpetuation. No man, I believe, is capable of sacri- 
ficing his life to any abstract idea ; in all cases it is the 
consciousness of contact with some being that enables 



Higher Space and Higher Being. 71 

him to make the last human sacrifice. And what we 
can do by this study of higher space, is to make this 
consciousness, which has been reserved for a few, .the 
property of all. Do we not all feel that there is a limit 
to our devotion to abstractions, none to beings whom 
we love. And to love them, we must know them. 

Then, just as our own individual life is empty and 
meaningless without those we love, so the life of the 
human race is empty and meaningless without a know- 
ledge of those that surround it. And although to some 
an inner knowledge of the oneness of all men is vouch- 
safed, it remains to be demonstrated to the many. 

The perpetual struggle between individual interests 
and the common good can only be solved by merging 
both impulses in a love towards one being whose life 
lies in the fulfilment of each. 

And this search, it seems to me, affords the needful 
supplement to the inquiries of one with whose thought 
I have been very familiar, and to which I return again, 
after having abandoned it for the purely materialistic 
views which seem forced upon us by the facts of science. 

All that he said seemed to me unsupported by fact, 
unrelated to what we know. 

But when I found that my knowledge was merely an 
empty pretence, that it was the vanity of being able to 
predict and foretell that stood to me in the place of an 
absolute apprehension of fact — when all my intellectual 
possessions turned to nothingness, then I was forced 
into that simple quest for fact, which, when persisted in 
and lived in, opens out to the thoughts like a flower to 
the life-giving sun. 

It is indeed a far safer course, to believe that which 
appeals to us as noble, than simply to ask what is true ; 
to take that which great minds have given, than to de- 
mand that our puny ones should be satisfied. But I 



72 A New Era of Thought. 

suppose there is some good to some one in the scep- 
ticism and struggle of those who cannot follow in the 
safer course. 

The thoughts of the inquirer to whom I allude may 
roughly be stated thus : — 

He saw in human life the working out of a great pro- 
cess, in the toil and strain of our human history he saw 
the becoming of man. There is a defect whereby we fall 
short of the true measure of our being, and that defect 
is made good in the course of history. 

It is owing to that defect that we perceive evil ; and 
in the perception of evil and suffering lies our healing, 
for we shall be forced into that path at last, after trying 
every other, which is the true one. 

And this, the history of the redemption of man, is 
what he saw in all the scenes of life ; each most trivial 
occurrence was great and significant in relation to this. 

And, further, he put forward a definite statement with 
regard to this defect, this lack of true being, for it lay, 
he said, in the self-centredness of our emotions, in the 
limitation of them to our bodily selves. He looked for 
a time when, driven from all thoughts of our own pain 
or pleasure, good or evil, we should say, in view of the 
miseries of our fellow-creatures, Let me be anyhow, use 
my body and my mind in any way, so that I serve. 

And this, it seems to me, is the true aspiration ; for, 
just as a note of music flings itself into the march of the 
melody, and, losing itself in it, is used for it and lost as 
a separate being, so we should throw these lives of ours 
as freely into the service of — whom .'' 

Here comes the difficulty. Let it be granted that we 
should have no self-rights, limit our service in no way, 
still the question comes, What shall we serve } 

It is far happier to have some concrete object to 
which we are devoted, or to be bound up in the cease- 



Higher Space and Higher Being. 73 

less round of active life, wherein each day presents so 
many necessities that we have no room for choice. 

But besides and apart from all these, there comes to 
some the question, "What does it all mean?" To others, 
an unlovable and gloomy aspect is presented, wherein 
their life seems to be but used as a material worthless in 
itself and ungifted with any dignity or honour ; while 
to others again, with the love of those they love, comes 
a cessation of all personal interest in life, and a dis- 
appointment and feeling of valuelessness. 

And in all these cases some answer is needed. And 
here human duty ceases. We cannot make objects to 
love. We can make machines and works of art, but 
nothing which directly excites our love. To give us 
that which rouses our love, is the duty of one higher 
than ourselves. 

And yet in one respect we have a duty — we must 
look. 

What gqod would it be, to surround us with objects 
of loving interest, if we bury our regards in ourselves 
and will not see ? 

And does it not seem as if with lowered eyelids, till 
only the 'thinnest slit was open, we gazed persistently, 
not on what is, but on the thinnest conceivable section 
of it } 

Let it be granted that our right attitude is, so to 
devote ourselves that there is no question as to what we 
will do or what we will not do, but we are perfectly 
obedient servants. The question is, Whom are we to 
serve .'' 

It cannot be each individual, for their claims are 
conflicting, and as often as not there is more need of 
a master than of a servant. Moreover, the aspect of our 
fellows does not always excite love, which is the only 
possible inducer of the right attitude of service. If we 



74 A New Era of Thought. 

do not love, we can only serve for a self motive, because 
it is in some way good for ourselves. 

Thus it seems to me that we are reduced to this : our 
only duty is to look for that which it is given us to love. 

But this looking is not mere gazing. To know, we 
must act. 

Let any one try it. He will find that unless he 
goes through a series of actions corresponding to his 
knowledge, he gets merely a theoretic and outside view 
of any facts. The way to know is this : Get somehow 
a means of telling what your perceptions would be if 
you knew, and act in accordance with those perceptions. 

Thus, with regard to a fellow-creature, if we knew him 
we should feel what his feelings are. Let us then learn 
his feelings, and act as if we had them. It is by the 
practical work of satisfying his needs that we get to 
know him. 

Then, may-be, we love him ; or perchance it is said 
we may find that through him we have been brought 
into contact with one greater than him. 

This is our duty— to know — to know, not merely 
theoretically, but practically ; and then, when we know, 
we have done our part ; if there is nothing, we cannot 
supply it. All we have to do is to look for realities. 

We must not take this view of education — that we are 
horribly pressed for time, and must learn, somehow, a 
knack of saying how things must be, without looking at 
them. 

But rather, we must say that we have a long time — all 
our lives, in which we will press facts closer and closer 
to our minds ; and we begin by learning the simplest. 
There is an idea in that home of our inspiration — the 
fact that there are certain mechanical processes by 
which men can acquire merit. This is perfectly true. 
It is by mechanical processes that we become different ; 



Perception and Inspiration. 75 

and the science of education consists largely in sys- 
tematizing these processes. 

Then, just as space perceptions are necessary for the 
knowledge of our fellow-men, and enable us to enter 
into human relationships with them in all the organized 
variety of civilized life, so it is necessary to develop 
our perceptions of higher space, so that we can appre- 
hend with our minds the relationship which we have to 
beings higher than ourselves, and bring our instinctive 
knowledge into clearer consciousness. 

It appears to me self-evident, that in the particular 
disposition of any portion of matter, that is, in any 
physical action, there can be neither right nor wrong ; 
the thing done is perfectly indifferent. 

At the same time, it is only in things done that we 
come into relationship with the beings about us and 
higher than us. Consequently, in the things we do lies 
the whole importance of our lives. 

Now, many of our impulses are directly signs of a 
relationship in us to a being of which we are not imme- 
diately conscious. The feeling of love, for instance, is 
always directed towards a particular individual ; but by 
love man tends towards the preservation and improve- 
ment of his race ; thus in the commonest and most 
universal impulses lie his relations to higher beings than 
the individuals by whom he is surrounded. Now, along 
with these impulses are many instincts of a modifying 
tendency ; and, being altogether in the dark as to the 
nature of the higher beings to whom we are related, it is 
difficult to say in what the service of the higher beings 
consists, in what it does not. The only way is, as in 
every other pre-rational department of life, to take the 
verdict of those with the most insight and inspiration. 

And any striving against such verdicts, and discontent 
with them, should be turned into energy towards finding 



76 A New Era of Thought. 

out exactly what relation we have towards these higher 
beings by the study of Space. 

Human life at present is an art constructed in its 
regulations and rules on the inspirations of those who 
love the undiscerned higher beings, of which we are a 
part. They love these higher beings, and know their 
service. 

But our perceptions are coarser ; and it is only by 
labour and toil that we shall be brought also to see, and 
then lose the restraints that now are necessary to us in 
the fulness of love. 

Exactly what relationship there is towards us on the 
part of these higher beings we cannot say in the least. 
We cannot even say whether there is more than humanity 
before the highest ; and any conception which we form 
now must use the human drama as its only possible 
mode of presentation. 

But that there is such a relation seems clear ; and the 
ludicrous manner, in which our perceptions have been 
limited, is a sufficient explanation of why they have not 
been scientifically apprehended. 

The mode, in which an apprehension of these higher 
beings or being is at present secured, is as follows ; and 
it bears a striking analogy to the mode by which the 
self is cut out of a block of cubes. 

When we study a block of cubes, we first of all learn 
it, by starting from a particular cube, and learning how 
all the others come with regard to that. All the others 
are right or left, up or down, near or far, with regard to 
that particular cube. And the line of cubes starting 
from this first one, which we take as the direction in 
which we look, is, as it were, an axis about which the 
rest of the cubes are grouped. We learn the block with 
regard to this axis, so that we can mentally conceive 
the disposition of every cube as it comes regarded from 



Perception and Inspiration. 77 

one point of view. Next we suppose ourselves to be in 
another cube , at the extremity of another axis ; and, 
looking from this axis, we learn the aspects of all the 
cubes, and so on. 

Thus we impress on the feeling what the block of 
cubes is like from every axis. In this way we get a 
knowledge of the block of cubes. 

Now, to get a knowledge of humanity, we must feel 
with many individuals. Each individual is an axis as 
it were, and we must regard human beings from many 
different axes. And as, in learning the block of cubes, 
muscular action, as used in putting up the block of 
cubes, is the means by which we impress on the feeling 
the different views of the block ; so, with regard to 
humanity, it is by acting with regard to the view of each 
individual that a knowledge is obtained. That is to say, 
that, besides sympathizing with each individual, we must 
act with regard to his view ; and acting so, we shall feel 
his view, and thus get to know humanity from more than 
one axis. Thus there springs up a feeling of humanity, 
and of more. 

Those who feel superficially with a great many people, 
are like those learners who have a slight acquaintance 
with a block of cubes from many points of view. Those 
who have some deep attachments, are like those who 
know them well from one or two points of view. 

Thus there are two definite paths — one by which the 
instinctive feeling is called out and developed, the other 
by which we gain the faculty of rationally apprehending 
and learning the higher beings. 

In the one way it is by the exercise of a sympathetic 
and active life ; in the other, by the study of higher 
space. 

Both should be followed ; but the latter way is more 
accessible to those who are not good. For we at any 



78 A New Era of Thought. 

rate have the industry to go through mechanical opera- 
tions, and know that we need something. 

And after all, perhaps, the difference between the good 
and the rest of us, lies rather in the former being aware. 
There is something outside them which draws them to 
it, which they see while we do not. 

There is no reason, however, why this knowledge 
should not become demonstrable fact. Surely, it is only 
by becoming demonstrable fact that the errors which 
have been necessarily introduced into it by human 
weakness will fall away from it. 

The rational knowledge will not replace feeling, but 
will form the vehicle by which the facts will be presented 
to our consciousness. Just as we learn to know our 
fellows by watching their deeds, — but it is something 
beyond the mere power of observing them that makes 
us regard them, — so the higher existences need to be 
known ; and, when known, then there is a chance that 
in the depths of our nature they will awaken feelings 
towards them like the natural response of one human 
being to another. 

And when we reflect on what surrounds us, when we 
think that the beauty of fruit and flower, the blue depths 
of the sky, the majesty of rock and ocean, — all these are 
but the chance and arbitrary view which we have of true 
being, — then we can imagine somewhat of the glories that 
await our coming. How set out in exquisite loveliness 
are all the budding trees and hedgerows on a spring day 
— from here, where they almost sing to us in their near- 
ness, to where, in the distance, they stand up delicately 
distant and distinct in the amethyst ocean of the air ! 
And there, quiet and stately, revolve the slow moving 
.sun and the stars of the night. All. these are the frag- 
mentary views which we have of great beings to whom 
we are related, to whom we are linked, did we but realize 



Perception and Inspiration. 79 

it, by a bond of love and service in close connexions of 
mutual helpfulness. 

Just as here and there on the face of a woman sits the 
divine spirit of beauty, so that all cannot but love who 
look — so, presenting itself to us in all this mingled scene 
of air and ocean, plain and mountain, is a being of such 
loveliness that, did we but know with one accord in one 
stream, all our hearts would be carried in a perfect and 
willing service. It is not that we need to be made 
different ; we have but to look and gaze, and see that 
centre whereunto with joyful love all created beings 
move. 

But not with effortless wonder will our days be filled, 
but in toil and strong exertion ; for, just as now we all 
labour and strive for an object, our service is bound up 
with things which we do — so then we find no rest from 
labour, but the sense of solitude and isolation is gone. 
The bonds of brotherhood with our fellow-men grow 
strong, for we know one common purpose. And through 
the exquisite face of nature shines the spiritual light 
that gives us a great and never-failing comrade. 

Our task is a simple one — to lift from our mind that 
veil which somehow has fallen on us, to take that curious 
limitation from our perception, which at present is only 
transcended by inspiration. 

And the means to do it is by throwing aside our reason 
— by giving up the idea that what we think or are has 
any value. We too often sit as judges of nature, when 
all we can be are her humble learners. We have but 
to drink in of the inexhaustible fulness of being, press- 
ing it close into our minds, and letting our pride of being 
able to foretell vanish into dust. 

There is a curious passage in the works of Immanuel 

Kant,i in which he shows that space must be in the 

' The idea of space can "nicht aus den Verhaltnissen der 



8o A New Era of Thought. 

mind before we can observe things in space. " For," 
he says, " since everything we conceive is conceived as 
being in space, there is nothing which comes before our 
minds from which the idea of space can be derived ; 
it is equally present in the most rudimentary perception 
and the most complete." Hence he says that space 
belongs to the perceiving soul itself. Without going 
into this argument to abstract regions, it has a great 
amount of practical truth. All our perceptions are of 
things in space ; we cannot think of any detail, how- 
ever limited or isolated, which is not in space. 

Hence, in order to exercise our perceptive powers, 
it is well to have prepared beforehand a strong appre- 
hension of space and space relations. 

And so, as we pass on, is it not easily conceivable 
that, with our power of higher space perception so 
rudimentary and so unorganized, we should find it im- 
possible to perceive higher existences .' That mode of 
perception which it belongs to us to exercise is wanting. 
What wonder, then, that we cannot see the objects 
which are ready, were but our own part done 1 

Think how much has come into human life through 
exercising the power of the three-dimensional space 
perception, and we can form some measure, in a faint 
way, of what is in store for us. 

There is a certain reluctance in us in bringing any- 
thing, which before has been a matter of feeling, within 
the domain of conscious reason. We do not like to 
explain why the grass is green, flowers bright, and, 
above all, why we have the feelings which we pass 
through. 

But this objection and instinctive reluctance is chiefly 

ausseren Erscheinung durch Erfahrung erborgt sein, sondern diese 
aussere Erfahrung ist nur durch gedachte Vorstellung allererst 
moglich." 



Perception and Inspiration. 8i 

derived from the fact that explaining has got to mean 
explaining away. We so often think that a thing is 
explained, when it can be shown simply to be another 
form of something which we know already. And, in 
fact, the wearied mind often does long to have a 
phenomenon shown to be merely a deduction from 
certain known laws. 

But explanation proper is not of this kind ; it is 
introducing into the mind the new conception which 
is indicated by the phenomenon already present. 
Nature consists of many entities towards the appre- 
hension of which we strive. If for a time we break 
down the bounds which we have set up, and unify vast 
fields of observation under one common law, it is that 
the conceptions we formed at first are inadequate, and 
must be replaced by greater ones. But it is always 
the case, that, to understand nature, a conception must 
be formed in the mind. This process of growth in the 
mental history is hidden ; but it is the really important 
one. The new conception satisfies more facts than the 
old ones, is truer phenomenally ; and the arguments for 
it are its simplicity, its power of accounting for many 
facts. But the conception has to be formed first. And 
the real history of advance lies in the growth of the 
new conceptions which every now and then come to 
light. 

When the weather-wise savage looked at the sky at 
night, he saw many specks of yellow light, like' fire- 
flies, sprinkled amidst whitish fleece ; and sometimes 
the fleece remained, the fire-spots went, and rain came ; 
sometimes the fire-spots remained, and the night iVas 
fine. He did not see that the fire-points were ever .the 
same, the clouds different; but by feeling dimly, he 
knew enough for his purpose. 

But when the thinking mind turned itself on these 

G 



82 A New Era of Thought. 

appearances, there sprang up, — not all at once, but 
gradually, — the knowledge of the sublime existences of 
the distant heavens, and all the lore of the marvellous 
forms of water, of air, and the movements of the earth. 
Surely these realities, in which lies a wealth of em- 
bodied poetry, are well worth the delighted sensuous 
apprehension of the savage as he gazed. 

Perhaps something is lost, but in the realities, of 
which we know, there is compensation. And so, when 
we learn to understand the meaning of these mysterious 
changes, this course of natural events, we shall find in 
the greater realities amongst which we move a fair 
exchange for the instinctive reverence, which they now 
awaken in us. 

In this book the task is taken up of forming the 
most simple and elementary of the great conceptions 
that are about us. In the works of the poets, and still 
more in the pages of religious thinkers, lies an untold 
wealth of conception, the organization of which in our 
every-day intellectual life is the work of the practical 
educator. 

But none is capable of such simple demonstration 
and absolute presentation as this of higher space, and 
none so immediately opens our eyes to see the world 
as a different place. And, indeed, it is very instructive ; 
for when the new conception is formed, it is found to 
be quite simple and natural. We ask ourselves what 
we have gained ; and we answer : Nothing ; we have 
simply removed an obvious limitation. 

And this is universally true ; it is not that we must 
rise to the higher by a long and laborious process. We 
may have a long and laborious process to go through, 
but, when we find the higher, it is this : we discover our 
true selves, our essential being, the fact of our lives. 
In this case, we pass from the ridiculous limitation, to 



Jr'erceptxon ana inspiration. 83 

which our eyes and hands seem to be subject, of acting 
in a mere section of space, to the fuller knowledge and 
feeling of space as it is. How do we pass to this truer 
intellectual life ? Simply by observing, by laying aside 
our intellectual powers, and by looking at what is. 

We take that which is easiest to observe, not that 
which is easiest to define ; we take that which is the 
most definitely limited real thing, and use it as our 
touchstone whereby to explore nature. 

As it seems to me, Kant made the great and funda- 
mental statement in philosophy when he exploded all 
previous systems, and all physics were reft from off the 
perceiving soul. But what he did once and for afl, was too 
great to be a practical means of intellectual work; The 
dynamic form of his absolute insight had to be found ; 
and it is in other works that the practical instances 
of the Kantian method are to be found. For, instead 
of looking at the large foundations of knowledge, the 
ultimate principles of experience, late writers turned 
to the details of experience, and tested every pheno- 
menon, not with the question. What is this ? but with 
the question, " What makes me perceive thus ? " 

And surely the question, as so put, is more capable 
of an answer ; for it is only the percipient, as a subject 
of thought, about which we can speak. The absolute 
soul, since it is the thinker, can never be the subject of 
thought ; but, as physically conditioned, it can be thought 
about. Thus we can never, without committing a 
ludicrous error, think of the mind of man except as 
a material organ of some kind ; and the path of dis- 
covery lies in investigating what the devious line of his' 
thought history is due to, which winds between two 
domains of physics — the unknown conditions which 
affect the perceiver, the partially known physics 
which constitute what we call the external world. 



84 A New Era of Thought. 

It is a pity to spend time over these reflections ; 
if they do not seem tame and poor compared to the 
practical apprehension which comes of working with 
the models, then there is nothing in the whole subject. 
If in the little real objects which the reader has to 
handle and observe does not lie to him a poetry of a 
higher kind than any expressed thought, then all these 
words are not only useless, but false. If, on the other 
hand, there is true work to be done with them, then 
these suggestions will be felt to be but mean and 
insufficient apprehensions. * 

For, in the simplest apprehension of a higher space 
lies a knowledge of a reality which is, to the realities 
we know, as spirit is to matter ; and yet to this new 
vision all our solid facts and material conditions are 
but as a shadow is to that which casts it. In the 
awakening light of this new apprehension, the flimsy 
world quivers and shakes, rigid solids flow and mingle, 
all our material limitations turn into graciousness, and 
the new field of possibility waits for us to look and 
behold. 



CHAPTER XI. 

SPACE THE SCIENTIFIC BASIS, OF ALTRUISM AND 
RELIGION. 

The reader will doubtless ask for some definite result 
corresponding to these words — something not of the 
nature of an hypothesis or a might-be. And in that I 
can only satisfy him after my own powers. My only 
strength is in detail and patience ; and if he will go 
through the practical part of the book, it will assuredly 
dawn upon him that here is the beginning of an answer 
to his request. I only study the blocks and stones 
of the higher life. But here they are definite enough. 
And the more eager he is for personal and spiritual truth, 
the more eagerly do I urge him to take up the practical 
work, for the true good comes to us through those who, 
aspiring greatly, still submit their aspirations to fact, 
and who, desiring to apprehend spirit, still are willing to 
manipulate matter. 

The particular problem at which I have worked for 
more than ten years, has been completely solved. It is 
possible for the mind to acquire a conception of higher 
space as adequate as that of our three-dimensional 
space, and to use it in the same manner. 

There are two distinct ways of studying space — our 
familiar space at present in use. One is that of the 
analyst, who treats space relations by his algebra, and 
discovers marvellous relations. The other is that of the 
observer or mechanician, who studies the shapes of things 
in space directly. 

8s 



86 A New Era of Thought. 

A practical designer of machines would not find the 
knowledge of geometrical analysis of immediate help to 
him ; and an artist or draughtsman still less so. 

Now, my inquiry was, whether it was possible to get 
the same power of conception of four-dimensional space, 
as the designer and draughtsman have of three-dimen- 
sional space. It is possible. 

And with this power it is possible for us to design 
machines in higher space, and to conceive objects in 
this space, just as a draughtsman or artist does. 

Analytical skill is not of much use in designing a 
statue or inventing a machine, or in appreciating the 
detail of either a work of art or a mechanical con- 
trivance. 

And hitherto the study of four-dimensional space has 
been conducted by analysis. Here, for the first time, 
the fact of the power of conception of four-dimensional 
space is demonstrated, and the means of educating it 
are given. 

And I propose a complete system of work, of which 
the volume on four space ^ is the first instalment. 

I shall bring forward a complete system of four- 
dimensional thought — mechanics, science, and art. The 
necessary condition is, that the mind acquire the power 
of using four-dimensional space as it now does three- 
dimensional. 

And there is another condition which is no less im- 
portant. We can never see, for instance, four-di- 
mensional pictures with our bodily eyes, but we can 
with our mental and inner eye. The condition is, that 
we should acquire the power of mentally carrying a 
great number of details. 

If, for instance, we could think of the human body 

' " Science Romance," No. I., by C. H. Hinton. Published by 
Swan Sonnenschein & Co. 



space the Basis of Altruism and Religion. 87 

right down to every minute part in its right position, 
and conceive its aspect, we should have a four-di- 
mensional picture which is a solid structure. Now, to 
do this, we must form the habit of mental painting, that 
is, of putting definite colours in definite positions, not 
with our hands on paper, but with our minds in thought, 
so that we can recall, alter, and view complicated arrange- 
ments of colour existing in thought with the same ease 
with which we can paint on canvas. This is simply an 
affair of industry ; and the mental power latent in us in 
this direction is simply marvellous. 

In any picture, a stroke of the brush put on without 
thought is valueless. The artist is not conscious of the 
thought process he goes through. For our purpose it 
is necessary that the manipulation of colour and form 
which the artist goes through unconsciously, should be- 
come a conscious power, and that, at whatever sacrifice 
of immediate beauty, the art of mental painting should 
exist beside our more unconscious art. All that I mean 
is this — that in the course of our campaign it is necessary 
to take up the task of learning pictures by heart, so 
that, just as an artist thinks over the outlines of a figure 
he wants to draw, so we think over each stroke in our 
pictures. The means by which this can be done will be 
given in a future volume. 

We throw ourselves on an enterprise in which we have 
to leave altogether the direct presentation to the senses. 
We must acquire a sense-perception and memory of so 
keen and accurate a kind that we can build up mental 
pictures of greater complexity than any which we can 
see. We have a vast work of organization, but it is 
merely organization. The power really exists and 
shows itself when it is looked for. 

Much fault may be found with the system of organi- 
zation which I have adopted, but it is the survivor oj" 



88 A New Era of Thought. 

many attempts ; and although I could better it in parts, 
still I think it is best to use it until, the full importance 
of the subject being' realized, it will be the lifework of 
men of science to reorganize the methods. 

The one thing on which I must insist is this — that 
knowledge is of no value, it does not exist unless it 
comes into the mind. To know that a thing must be is 
no use at all. It must be clearly realized, and in detail 
as it is, before it can be used. 

A whole world swims before us, the apprehension 
of which simply demands a patient cultivation of our 
powers ; and then, when the faculty is formed, we shall 
recognize what the universe in which we are is like. We 
shall learn about ourselves and pass into a new domain. 

And I would speak to some minds who, like myself, 
share to a large extent the feeling of unsettledness and 
unfixedness of our present knowledge. 

Religion has suffered in some respects from the in- 
accuracy of its statements ; and it is not always seen 
that it consists of two parts — one a set of rules as to the 
management of our relations to the physical world about 
us, and to our own bodies ; another, a set of rules as to 
our relationship to beings higher than ourselves. 

Now, on the former of these subjects, on physical facts, 
on the laws of health, science has a fair standing ground 
of criticism, and can correct the religious doctrines in 
many important respects. 

But on the other part of the subject matter, as to our 
relationship to beings higher than ourselves, science 
has not yet the materials for judging. The proposition 
which underlies this book is, that we should begin to 
acquire the faculties for judging. 

To judge, we must first appreciate ; and how far we 
are from appreciating with science the fundamental 
religious doctrines I leave to any one to judge. 



space the Basis of Altruism and Religion. 89 

There is absolutely no scientific basis for morality, 
using morality in the higher sense of other than a code 
of rules to promote the greatest physical and mental 
health and growth of a human being. Science does not 
give us any information which is not equally acceptable 
to the most selfish and most generous man ; it simply 
tells him of means by which he may attain his own 
ends, it does not show him ends. 

The prosecution of science is an ennobling pursuit ; 
but it is of scientific knowledge that I am now speaking 
in itself. We have no scientific knowledge of any exist- 
ences higher than ourselves — at least, not recognized 
as higher. But we have abundant knowledge of the 
actions of beings less developed than ourselves, from 
the striking unanimity with which all inorganic beings 
tend to move towards the earth's centre, to the almost 
equally uniform modes of response in elementary or- 
ganized matter to different stimuli. 

The question may be put : In what way do we come 
into contact with these higher beings at present .' And 
evidently the answer is. In those ways in which we 
tend to form organic unions — unions in which the activi- 
ties of individuals coalesce in a living way. 

The coherence of a military empire or of a subju- 
gated population, presenting no natural nucleus of 
growth, is not one through which we should hope 
to grow into direct contact with our higher destinies. 
But in friendship, in voluntary associations, and above 
all, in the family, we tend towards our greater life. 

And it seems that the instincts of women are much 
more relative to this, the most fundamental and import- 
ant side of life, than are those of men. In fact, until 
we know, the line of advance had better be left to the 
feeling of women, as they organize the home and the 
social life spreading out therefrom. It is difficult, perhaps, 



90 A New Era of Thought. 

for a man to be still and perceive ; but if he is so, he 
finds that what, when thwarted, are meaningless caprices 
and empty emotionalities, are, on the part of woman, 
when allowed to grow freely and unchecked, the first 
beginnings of a new life — the shadowy filaments, as it 
were, by which an organism begins to coagulate to- 
gether from the medium in which it makes its appearance. 

In very many respects men have to make the condi- 
tions, and then learn to recognize. How can we see 
the higher beings about us, when we cannot even 
conceive the simplest higher shapes ? We may talk 
about space, and use big words, but; after all, the prefer- 
able way of putting our efforts is this : let us look first 
at the simplest facts of higher existence, and then, when 
we have learnt to realize these. We shall be able to see 
what the world presents. And then, also, light will be 
thrown on the constituent organisms of our own bodies, 
when we see in the thorough development of our social 
life a relation between ourselves and a larger organism 
similar to that which exists between us and the minute 
constituents of oUr frame. 

The problem, as it comes to me, is this : it is clearly 
demonstrated that self-regard is to be put on one side — 
and self-regard in every respect — not only should things 
painful and arduous be done, but things degrading and 
vile, so that they serve. 

I am to sign any list of any number of deeds which 
the most foul imagination can suggest, as things which I 
would do did the occasion come when I could benefit 
another by doing them ; and, in fact, there is to be no 
characteristic in any action v/hich I would shrink from 
did the occasion come when it presented itself to be 
done for another's sake. And I believe that the soul 
is absolutely unstained by the action, provided the re- 
gard is for another. 



space the Basis of Altruism and Religion. 91 

But this is, in truth, a dangerous doctrine ; at one 
Sweep it puts away all absolute commandments, all 
absolute verdicts of right about things, and leaves the 
agent to his own judgment. 

It is a kind of rule of life which requires most abso- 
lute openness, and demands that society should frame 
severe and insuperable regulations ; for otherwise, with 
the motives of the individual thus liberated from absolute 
law, endless varieties of conduct would spring forth, 
and the wisdom of individual men is hardly enough to 
justify their irresponsible action. 

Still, it does seem that, as an ideal, the absolute 
absence of self-regard is to be aimed at. 

With a strong religious basis, this would work no 
harm, for the rules of life, as laid down by religions, 
would suffice. But there are many who do not accept 
these rules as any absolute indication of the will of 
God, but only as the regulations of good men, which 
have a claim to respect and nothing more. 

And thus it seems to me that altruism — thorough- 
going altruism — hands over those who regard it as an 
ideal, and who are also of a sceptical turn of mind, to 
the most absolute unfixedness of theory, and, very pos- 
sibly, to the greatest errors in life. 

And here we come to the point where the study of 
space becomes so important. 

For if this rule of altruism is the right one, if it 
appeals with a great invitation to us, we need not there- 
fore try it with less precaution than we should use in 
other affairs of infinitely less importance. When we 
want to know if a plank will bear, we entrust it with 
a different load from that of a human body. 

And if this law of altruism is the true one, let us try 
it where failure will not mean the ruin of human 
beings. 



^2 A New Era of Thought. 

Now, in knowledge, pure altruism means so to bury 
the mind in the thing known that all particular relations 
of one's self pass away. The altruistic knowledge of 
the heavens would be, to feel that the stars were vast 
bodies, and that I am moving rapidly. It would be, to 
know this, not as a matter of theory, but as a matter 
of habitual feeling. 

Whether this is possible, I do not know ; but a some- 
what similar attempt can be made with much simpler 
means. 

In a different place I have described the process of 
acquiring an altruistic knowledge of a block of cubes ; 
and the results of the laborious processes involved are 
well worth the trouble. For as a clearly demonstrable 
fact this comes before one. To acquire an absolute 
knowledge of a block of cubes, so that all self relations 
are cast out, means that one has to take the view of a 
higher being. 

It suddenly comes before one, that the particular re- 
lations which are so fixed and important, and seem so 
absolutely sure when one begins the process of learning, 
are by no means absolute facts, but marks of a singular 
limitation, almost a degradation, on one's own part. In 
the determined attempt to know the most insignificant 
object perfectly and thoroughly, there flashes before 
one's eyes an existence infinitely higher than one's own. 
And with that vision there comes, — I do not speak 
from my own experience only, — a conviction that our 
existence also is not what we suppose — that this 
bodily self of ours is but a limit too. And the question 
of altruism, as against self-regard, seems almost to 
vanish, for by altruism we come to know what we truly 
are. 

" What we truly are," I do not mean apart from space 
and matter, but what we really are as beings having a 



space the Basis of Altruism and Religion. 93 

space existence ; for our way of thinking about existence 
is to conceive it as the relations of bodies in space. To 
think is to conceive realities in space. 

Just as, to explore the distant stars of the heavens, a 
particular material arrangement is necessary which we 
call a telescope, so to explore the nature of the beings 
who are higher than us, a mental arrangement is ne- 
cessary. We must prepare our power of thinking as 
we prepare a more extended power of looking. We 
want a structure developed inside the skull for the 
one purpose, while an exterior telescope will do for the 
other. 

And thus it seems that the difficulties which we first 
apprehended fall away. 

To us, looking with half-blinded eyes at merely our 
own little slice of existence, our filmy all, it seemed 
that altruism meant disorder, vagary, danger. 

But when we put it into practice in knowledge, we 
find that it means the direct revelation of a higher 
being and a call to us to participate ourselves too in a 
higher life — nay, a consciousness comes that we are 
higher than we know. 

And so with our moral life as with our intellectual 
life. Is it not the case that those, who truly accept the 
rule of altruism, learn life in new dangerous ways ? 

It is true that we must give up the precepts of religion 
as being the will of God ; but then we shall learn that 
the will of God shows itself partly in the religious pre- 
cepts, and comes to be more fully and more plainly 
known as an inward spirit. 

And that difficulty, too, about what we may do and 
what we may not, vanishes also. For, if it is the same 
about our fellow-creatures as it is about the block of 
cubes, when we have thrown out the self-regard from 
our relationship to them, we shall feel towards them as 



94 A New Era of Thought. 

a higher being than man feels towards them, we shall 
feel towards them as they are in their true selves, not in 
their outward forms, but as eternal loving spirits. 

And then those instincts which humanity feels with a 
secret impulse to be sacred and higher than any tem- 
porary good will be justified — or fulfilled. 

There are two tendencies — one towards the direct 
cultivation of the religious perceptions, the other to re- 
ducing everything to reason. It will be but just for the 
exponents of the latter tendency to look at the whole 
universe, not the mere section of it which we know, be- 
fore they deal authoritatively with the higher parts of 
religion. 

And those who feel the immanence of a higher life in 
us will be needed in this outlook on the wider field of 
reality, so that they, being fitted to recognize, may tell 
us what lies ready for us to know. 

The true path of wisdom consists in seeing that our 
intellect is foolishness — that our conclusions are absurd 
and mistaken, not in speculating on the world as a form 
of thought projected from the thinking principle within 
us — rather to be amazed that our thought has so limited 
the world and hidden from us its real existences. To 
think of ourselves as any other than things in space and 
subject to material conditions, is absurd, it is absurd on 
either of two hypotheses. If we are really things in 
space, then of course it is absurd to think of ourselves 
as if we were not so. On the other hand, if we are not 
things in space, then conceiving in space is the mode 
in which that unknown which we are exists as a mind. 
Its mental action is space-conception, and then to give 
up the idea of ourselves as in space, is not to get a truer 
idea, but to lose the only power of apprehension of our- 
selves which we possess. 

And yet there is, it must be confessed, one way in 



space the Bans of Altruism and Religion. 95 

which it may be possible for us to think without think- 
ing of things in space. 

That way is, not to abandon the use of space-thought, 
but to pass through it. 

When we think of space, we have to think of it as in- 
finity extended, and we have to think of it as of infinite 
dimensions. Now, as I have shown in " The Law of the 
Valley," ^ when we come upon infinity in any mode of 
our thought, it is a sign that that mode of thought is 
dealing with a higher reality than it is adapted for, and 
in struggling to represent it, can only do so by an in- 
finite number of terms. Now, space has an infinite 
number of positions and turns, and this may be due to 
the attempt forced upon us to think of things higher 
than space as in space. If so, then the way to get rid 
of space from our thoughts, is, not to go away from it, 
but to pass through it — to think about larger and larger 
systems of space, and space of more and more dimen- 
sions, till at last we get to such a representation in 
space of what is higher than space, that we can pass 
from the space-thought to the more absolute thought 
without that leap which would be necessary if we were 
to try to pass beyond space with our present very in- 
adequate representation in it of what really is. 

Again and again has human nature aspired and 
fallen. The vision has presented itself of a law which 
was love, a duty which carried away the enthusiasm, 
and in which the conflict of the higher and lower natures 
ceased because all was enlisted in one loving service. 
But again and again have such attempts failed. The 
common-sense view, that man is subject to law, external 
law, remains — that there are fates whom he must pro- 
pitiate and obey. And there is a strong sharp curb, 

1 " Science Romances," No. II. 



96 A New Era of Thought. 

which, if it be not brought to bear by the will, is soon 
pulled tight by the world, and one more tragedy is 
enacted, and the over-confident soul is brought low. 

And the rock on which such attempts always split, 
is in the indulgence of some limited passion. Some 
one object fills the soul with its image, and in devotion 
to that, other things are sacrificed, until at last all 
comes to ruin. 

But what does this mean > Surely it is simply this, 
that where there should be knowledge there is ignorance. 
It is not that there is too much devotion, too much 
passion, but that we are ignorant and blind, and 
wander in error. We do not know what it is we care 
for, and waste our effort on the appearance. There is 
no such thing as wrong love ; there is good love and 
bad knowledge, and men who err, clasp phantoms to 
themselves. Religion is but the search for realities ; 
and thought, conscious of its own limitations, is its 
best aid. 

■ Let a man care for any one object — let his regard 
for it be as concentrated and exclusive as you will, 
there will be no danger if he truly apprehends that 
which he cares for. Its true being is bound up with 
all the rest of existence, and, if his regard is true to 
one, then, if that one is really known, his regard is 
true to all. 

There is a question sometimes asked, which shows 
the mere formalism into which we have fallen. 

We ask : What is the end of existence .' A mere play 
on words ! For to conceive existence is to feel ends. 
The knowledge of existence is the caring for objects, 
the fear of dangers, the anxieties of love. Immersed 
in these, the triviality of the question, what is the end of 
existence ? becomes obvious. If, however, letting reality 
fade away, we play with words, some questions of this 



space the Basis of Altruism and Religion. 97 

kind are possible ; but they are mere questions of words, 
and all content and meaning has passed out of them. 

The task before us is this : we strive to find out that 
physical unity, that body which men are parts of, and 
in the life of which their true -unity lies. The existence 
of this one body we know from the utterances of those 
whom we cannot but feel to be inspired ; we feel certain 
tendencies in ourselves which cannot be explained 
except by a supposition of this kind. 

And, now, we set to work deliberately to form in 
our minds the means of investigation, the faculty of 
higher-space conception. To our ordinary space- 
thought, men are isolated, distinct, in great measure 
antagonistic. But with the first use of the weapon of 
higher thought, it is easily seen that all men may really 
be members of one body, their isolation may be but an 
affair of limited consciousness. There is, of course, no 
value as science in such a supposition. But it suggests 
to us many possibilities ; it reveals to us the confined 
nature of our present physical views, and stimulates us 
to undertake the work necessary to enable us to deal 
adequately with the subject. 

The work is entirely practical and detailed ; it is the 
elaboration, beginning from the simplest objects of an 
experience in thought, of a higher-space world. 

To begin it, we take up those details of position and 
relation which are generally relegated to symbolism or 
unconscious apprehension, and bring these waste pro- 
ducts of thought into the central position of the labora- 
tory of the mind. We turn all our attention on the 
most simple and obvious details of our every-day ex- 
perience, and thence we build up a conception of the 
fundamental facts of position and arrangement in a 
higher world. We next study more complicated higher 
shapes, and get our space perception drilled and dis- 

H 



98 New Era of Thought, 

ciplined. Then we proceed to put a content into our 
framework. 

The means of doing this are twofold — observation 
and inspiration. 

As to observation, it is hardly possible to describe 
the feelings of that; investigator who shall distinctly 
trace in the physical world, and experimentally de- 
monstrate the existence of the higher-space facts which 
are so curiously hidden from us. He will lay the first 
stone for the observation and knowledge of the higher 
beings to whom we are related. 

As to the other means, it is obvious, surely, that if 
there has ever been inspiration, there is inspiration 
now. Inspiration is not a unique phenomenon. It has 
existed in absolutely marvellous degree in some of the 
teachers of the ancient world ; but that, whatever it 
was, which they possessed, must be present now, and, 
if we could isolate it, be a demonstrable fact. 

And I would propose to define inspiration as the 
faculty, which, to take a particular instance, does the 
following '.-^ 

If a square penetrates a line cornerwise, it marks 
out on the line a segment bounded by two points — that 
is, we suppose a line drawn on a piece of paper, and 
a square lying on the paper to be pushed so that its 
corner passes over the line. Then, supposing the paper 
and the line to be in the same plane, the line is inter- 
rupted by the square ; and, of the square, all that is 
observable in the line, is a segment bounded by two 
points. 

Next, suppose a cube to be pushed cornerwise 
through a plane, and let the plane make a section of 
the cube. The section will be a plane figure, and it 
will be a triangle. 

Now, first, the section of a square by a line is a 



space the Basis of Altruism and Religion. 99 

segment bounded by two points ; second, the section 
of a cube by a plane is a triangle bounded by three 
lines. 

Hence, we infer that the section of a figure in four 
dimensions analogous to a cube, by three-dimensional 
space, will be a tetrahedron — a figure bounded by four 
planes. 

This is found to be true ; with a little familiarity 
with four-dimensional movements this is seen to be 
obvious. But I would define inspiration as the faculty 
by which without actual experience this conclusion 
is formed. 

How it is we come to this conclusion I am perfectly 
unable to say. Somehow, looking at mere formal con- 
siderations, there comes into the mind a conclusion 
about something beyond the range of actual experience. 

We may call this reasoning from analogy ; but using 
this phrase does not explain the process. It seems to 
me just as rational to say that the facts of the line and 
plane remind us of facts which we know already about 
four-dimensional figures — that they tend to bring these 
facts out into consciousness, as Plato shows with the 
boy's knowledge of the cube. We must be really four- 
diniensional creatures, or we could not think about four 
dimensions. 

But whatever name we give to this peculiar and in- 
explicable faculty, that we do possess it is certain ; and 
in our investigations it will be of service to us. We 
must carefully investigate existence in a plane world, 
and then, making sure, and impressing on our inward 
sense, as we go, every step we take with regard to a 
higher world, we shall be reminded continually of fresh 
possibilities of our higher existence. 



PART IL 
CHAPTER I. 

THREE- SPACE. GENESIS OF A CUBE. APPEARANCES 
OF A CUBE TO A PLANE-BEING. 

The models consist of a set of eight and a set of four 
cubes. They are marked with different colours, so as 
to show the properties of the figure in Higher Space, to 
which they belong. 

The simplest figure in one-dimensional space, that is, 
in a straight line, is a straight line bounded at the two 
extremities. The figure in this case consists of a length 
bounded by two points. 

Looking at Cube i, and placing it so that the figure i 
is uppermost, we notice a straight line in contact with 
the table, which is coloured Orange. It begins in a 
Gold point and ends in a Fawn point. The Orange 
extends to some distance on two faces of the Cube ; but 
for our present purpose we suppose it to be simply a 
thin line. 

This line we conceive to be generated in the following 
way. Let a point move and trace out a line. Let the 
point be the Gold point, and let it, moving, trace out the 
Orange line and terminate in the Fawn point. Thus 
the figure consists of the point at which it begins, the 
point at which it ends, and the portion between. We 
may suppose the point to start as a Gold point, to 



I02 New Era of Thought. 

change its colour to Orange during the motion, and 
when it stops to become Fawn. The motion we suppose 
from left to right, and its direction we call X. 

If, now, this Orange line move away from us at right 
angles, it will trace out a square. Let this be the Black 
square, which is seen underneath Model i. The points, 
which bound the line, will during this motion trace out 
lines, and to these lines there will be terminal points. 
Also, the Square will be terminated by a line on the 
opposite side. Let the Gold point in moving away 
trace out a Blue line and end in a Buff point ; the Fawn 
point a Crimson line ending in a Terracotta point. 
The Orange line, having traced a Black square, ends in 
a Green-grey line. This direction, away from the 
observer, we call Y. 

Now, let the whole Black square traced out by the 
Orange line move upwards at right angles. It will 
trace out a new figure, a Cube. And the edges of the 
square, while moving upwards, will traCe out squares. 
Bounding the cube, and opposite to the Black square, 
will be another square. Let the Orange line moving 
upwards trace a Dark Blue square and end in a Reddish 
line. The Gold point traces a Brown line ; the Fawn point 
traces a French-grey line, and these lines end in a Light- 
blue and a Dull-purple point. Let the Blue line trace a 
Vermilion square and end in a Deep-yellow line. Let 
the Buff point trace a Green line, and end in a Red 
point. The Green-grey line' traces a Light-yellow 
square and ends in a Leaden line ; the Terracotta point 
traces a Dark-slate line and ends in a Deep-blue point 
The Crimson line traces a Blue-green square and ends 
in a Bright-blue line. 

Finally, the Black square traces a Cube, the colour of 
which is invisible, and ends in a white square. We 
suppose the colour of the cube to be a Light-buff. The 



Thre&Space. Genesis of a Cube. 103 

upward direction we call Z. Thus we say : The Gold 
point moved Z, traces a Brown line, and ends in a Light- 
blue point. 

We can now clearly realize and refer to each region 
of the cube by a colour. 

At the Gold point, lines from three directions meet, 
the X line Orange, the Y line Blue, the Z line Brown. 

Thus we began with a figure of one dimension, a line, 
we passed on to a figure of two dimensions, a square, 
and ended with a figure of three dimensions, a cube. 



The square represents a figure in two dimensions ; but 
if we want to realize what it is to a being in two 
dimensions, we must not look down on it. Such a view 
could not be taken by a plane-being. 

Let us suppose a being moving on the surface of the 
table and unable to rise from it. Let it not know that 
it is upon anything, but let it believe that the two 
directions and compounds of those two directions are all 
possible directions. Moreover, let it not ask the ques- 
tion : "On what am I supported.'" Let it see no reason 
for any such question, but simply call the smooth surface, 
along which it moves, Space. 

Such a being could not tell the colour of the square 
traced by the Orange line. The square would be 
bounded by the lines which surround it, and only by 
breaking through one of those lines could the plane- 
being discover the colour of the square. 

In trying to realize the experience of a plane-being 
it is best to suppose that its two dimensions are upwards 
and sideways, i.e., Z and X, because, if there be any 
matter in the plane-world, it will, like matter in the 
solid world, exert attractions and repulsions. The 
matter, like the beings, must be supposed, very thin, that 



104 New Era of Thought. 

is, of so slight thickness that it is quite unnoticed by the 
being. Now, if there be a very large mass of such 
matter lying on the table, and a plane-being be free 
to move about it, he will be attracted to it in every 
direction. "Towards this huge mass" would be 
"Down," and "Away from it " would be " Up," just as 
" Towards the earth " is to solid beings " Down," and 
" Away from it " is " Up," at whatever part of the globe 
they may be. Hence, if we want to realize a plane- 
being's feelings, we must keep the sense of up and down^ 
Therefore we must use the Z direction, arid it is more 
convenient to take Z and X than Z and Y. 

Any direction lying between these is said to be com- 
pounded of the two ; for, if we move slantwise for some 
distance, the point reached might have been also reached 
by going a certain distance X, and then a certain 
distance Z, or vice versd. 

Let us suppose the Orange line has moved Z, and 
traced the Dark-blue square ending in the Reddish line. 
If we now place a piece of stiff paper against the Dark- 
blue square, and suppose the plane-beings to move to 
and fro on. that surface of the paper, which touches 
the square, we shall have means of representing their 
experience. 

To obtain a more consistent view of their existence, 
let us suppose the piece of paper extended, so that it 
cuts through our earth and comes out at the antipodes, 
thus cutting the earth in two. Then suppose all the 
earth removed away, both hemispheres vanishing, and 
only a very thin layer of matter left upon the paper on 
that side which touches the Dark-blue square. This 
represents what the world would be to a plane-being. 

It is of some importance to get the notion of the 
directions in a plane-world, as great difficulty arises 
from our notions of up and down. We miss the right 



Three- space. Genesis of a Cube. 105 

analogy if we conceive of a plane-world without the 
conception of up and down. 

A good plan is, to use a slanting surface, a stiff card 
or book cover, so placed that it slopes upwards to the 
eye. Then gravity acts as two forces. It acts (i) as a 
force pressing all particles upon the slanting surface into 
it, and (2) as a force of gravity along the plane, making 
particles tend to slip down its incline. We may suppose 
that in a plane-world there are two such forces, one 
keeping the beings thereon to the plane, the other 
acting between bodies in it, and of such a nature that by 
virtue of it any large mass of plane-matter produces on 
small particles around it the same effects as the large 
mass of solid matter called our earth produces on small 
objects like our bodies situated around it. In both cases 
the larger draws the smaller to itself, and creates the 
sensations of up and down. 

If we hold the cube so that its Dark-blue side touches 
a sheet of paper held upwards to the eye, and if we 
then look straight down along the paper, confining our 
view to that which is in actual contact with the paper, 
we see the same view of the cube as a plane-being 
would get. We see a Light-blue point, a Reddish line, 
and a Dull-purple point. The plane-being only sees a 
line, just as we only see a square of the cube. 

The line where the paper rests on the table may be 
taken as representative of the surface of the plane- 
being's earth. It would be merely a line to him, but it 
would have the same property in relation to the plane- 
world, as a square has in relation to a solid world ; in 
neither case can the notion of what in the latter is 
termed solidity be quite excluded. If the plane-being 
broke through the line bounding his earth, he would find 
more matter beyond it. 

Let us now leave out of consideration the question of 



io6 New Era of Thought. 

"up and down" in a plane-world. Let us no longer 
consider it in the vertical, or ZX, position, but simply 
take the surface (XY) of the table as that which sup- 
ports a plane-world. Let us represent its inhabitants 
by thin pieces of paper, which are free to move over the 
surface of the table, but cannot rise from it. Also, let 
the thickness (i.e., height above the surface) of these 
beings be so small that they cannot discern it. Lastly 
let us premise there is no attraction in their world, so 
that they have not any up and down. 

Placing Cube i in front of us, let us now ask how a 
plane-being could apprehend such a cube. The Black 
face he could easily study. He would find it bounded 
by Gold point, Orange line. Fawn point. Crimson line, 
and so on. And he would discover it was Black by 
cutting through any of these lines and entering it. 
(This operation would be equivalent to the mining of a 
solid being). 

But of what came above the Black square he would 
be completely ignorant. Let us now suppose a square 
hole to be made in the table, so that the cube could 
pass through, and let the cube fit the opening so 
exactly that no trace of the cutting of the table be 
visible to the plane-being. If the cube began to pass 
through, it would seem to him simply to change, for of 
its motion he could not be aware, as he would not know 
the direction in which it moved. Let it pass down till 
the White square be just on a level with the surface of 
the table. The plane-being would then perceive a 
Light-blue point, a Reddish line, a Dull-purple point, a 
Bright-blue line, and so on. These would surround a 
White square, which belonged to the same body as that 
to which the Black square belonged. But in this body 
there would be a dimension, which was not in the 
square. Our upward directio^i would not be appre- 



Three-Space. Genesis of a Cube. 107 

hended by him directly. Motion from above down- 
wards would only be apprehended as a change in the 
figure before him. He would not say that he had before 
him different sections of a cube, but only a changing 
square. If he wanted to look at the upper square, he 
could only do so when the Black square had gone an 
inch below his plane. To study the upper square 
simultaneously with the lower, he would have to make 
a model of it, and then he could place it beside the 
lower one. 

Looking at the cube, we see that the Reddish line 
corresponds precisely to the Orange line, and the Deep- 
yellow to the Blue line. But if the plane-being had a 
model of the upper square, and placed it on the right- 
hand side of the Black square, the Deep-yellow line 
would come next to the Crimson line of the Black 
square. There would be a discontinuity about it. All 
that he could do would be to observe which part in the 
one square corresponded to which part in the other. 
Obviously too there lies something between the Black 
square and the White. 

The plane-being would notice that when a line moves 
in a direction not its own, it traces out a square. When 
the Orange line is moved away, it traces out the Black 
square. The conception of a new direction thus ob- 
tained, he would understand that the Orange line 
moving so would trace out a square, and the Blue line 
moving so would do the same. To us these squares 
are visible as wholes, the Dark-blue, and the Vermilion. 
To him they would be matters of verbal definition 
rather than ascertained facts. However, given that he 
had the experience of a cube being pushed through his 
plane, he would know there was some figure, whereof 
his square was part, which was bounded by his square 
on one side, and by a White square on another side. 



io8 New Era of Thought. 

We have supposed him to make models of these boun- 
daries, a Black square and a White square. The Black 
square, which is his solid matter, is only one boundary 
of a figure in Higher Space. 

But we can suppose the cube to be presented to him 
otherwise than by passing through his plane. It can be 
turned round the Orange Hne, in which case the Blue 
line goes out, and, after a time, the Brown line comes 
in. It must be noticed that the Brown line comes into 
a direction opposite to that in which the Blue line 
ran. These two lines are at right angles to each other, 
and, if one be moved upwards till it is at right angles to 
the surface of the table, the other comes on to the sur- 
face, but runs in a direction opposite to that in which 
the first ran. Thus, by turning the cube about the 
Orange line and the Blue line, different sides of it can 
be shown to a plane-being. By combining the two 
processes of turning and pushing through the plane, all 
the sides can be shown to the plane-being. For in- 
stance, if the cube be turned so that the Dark-blue 
square be on the plane, and it be then passed through, 
the Light-yellow square will come in. 

Now, if the plane-being made a set of models of 
these different appearances and studied them, he could 
form some rational idea of the Higher Solid which 
produced them. He would become able to give some 
consistent account of the properties of this new kind 
of existence ; he could say what came into his plane 
space, if the other space penetrated the plane edge-wise 
or corner-wise, and could describe all that would come 
in as it turned about in any way. 

He would have six models. Let us consider two of 
them — the Black and the White squares. We can ob- 
serve them on the cube. Every colour on the one is 
different from every colour on the other. If we now 



Three-Space. Genesis of a Cube. 109 

ask what lies between the Orange line and the Reddish 
line, we know it is a square, for the Orange line moving 
in any direction gives a square. And, if the six models 
were before the plane-being, he could easily select that 
which showed what he wanted. For that which lies 
between Orange line and Reddish line must be bounded 
by Orange and Reddish lines. He would search among 
the six models lying beside each other on his plane, till 
he found the Dark-blue square. It is evident that only 
one other square differs fn all its colours from the Black 
square, viz., the White square. For it is entirely sepa- 
rate. The others meet it in one of their lines. This 
total difference exists in all the pairs of opposite sur- 
faces on the cube. 

Now, suppose the plane-being asked himself what 
would appear if the cube turned round the Blue line. 
The cube would begin to pass through his space. The 
Crimson line would disappear beneath the plane and 
the Blue-green square would cut it, so that opposite to 
the Blue line in the plane there would be a Blue-green 
line. The French-grey line and the Dark-slate line 
would be cut in points, and from the Gold point to the 
French-grey point would be a Dark-blue line ; and 
opposite to it would be a Light-yellow line, from the 
Buff point to the Dark-slate point. Thus the figure in 
the plane world would be an oblong instead of a square, 
and the interior of it would be of the same Light-buff 
colour as the interior of the cube. It is assumed that 
the plane closes up round the passing cube, as the sur- 
face of a liquid does round any object immersed. 

But, in order to apprehend what would take place 
when this twisting round the Blue line began, the plane- 
being would have to set to work by parts. He has no 
conception of what a solid would do in twisting, but 
he knows what a plane does. Let him, then, instead 



no New Era of Thought. 

of thinking of the whole Black square, think only of 
the Orange line. The Dark-blue square stands on it. 
As far as this square is concerned, twisting round the 
Blue line is the same as twisting round the Gold point. 
Let him imagine himself in that plane at right angles to 
his plane-world, which contains the Dark-blue square. 
Let him keep his attention fixed on the line where the 
two planes meet, viz., that which is at first marked by 
the Orange line. We will call this line the line of his 
plane, for all that he knows of his own plane is this 
line. Now, let the Dark-blue square turn round the 
Gold point. The Orange line at once dips below 
the line of his plane, and the Dark-blue square passes 
through it. Therefore, in his plane he will see a 
Dark-blue line in place of the Orange one. And in 
place of the Fawn point, only further off from the Gold 
point, will be a French -grey point. The Diagrams 
(i), (2) show how the cube appears as it is before and 
after the turning. G is the Gold, F the Fawn point. 
In (2) G is unmoved, and the plane is cut by the French- 
grey line, Gr. 

Instead of imagining a direction he did not know, the 
plane -being could think of the Dark-blue square as 
lying in his plane. But in this case the Black square 
would be out off his plane, and only the Orange line 
would remain in it. Diagram (3) shows the Dark-blue 
square lying in his plane, and Diagram (4) shows it 
turning round the Gold point. Here, instead of think- 
ing about his plane and also that at right angles to it, 
he has only to think how the square turning round the 
Gold point will cut the line, which runs left to right 
from G, viz., the dotted line. The French-grey line is 
cut by the dotted line in a point. To find out what 
would come in at other parts, he need only treat a 
number of the plane sections of the cube perpendicular 




LB LB 

a a 



\To face p. no. 



Three-Space. Genesis of a Cube. 1 1 1 

to the Black square in the same manner as he had 
treated the Dark-blue square. Every such section would 
turn round a point, as the whole cube turned round the 
Blue line. Thus he would treat the cube as a number 
of squares by taking parallel sections from the Dark- 
blue to the Light-yellow square, and he would turn 
each of these round a corner of the same colour as the 
Blue line. Combining these series of appearances, he 
would discover what came into his plane as the cube 
turned round the Blue line. Thus, the problem of the 
turning of the cube could be settled by the consideration 
of the turnings of a number of squares. 

As the cube turned, a number of different appear- 
ances would be presented to the plane - being. The 
Black square would change into a Light-buff oblong, 
with Dark-blue, Blue-green, Light-yellow, and Blue 
sides, and would gradually elongate itself until it be- 
came as long as the diagonal of the square side of 
the cube; and then the bounding line opposite to the 
Blue line would change from Blue-green to Bright-blue, 
the other lines remaining the same colour. If the cube 
then turned still further, the Bright-blue line would 
become White, and the oblong would diminish in length. 
It would in time become a Vermilion square, with a 
Deep-yellow line opposite to the Blue line. It would 
then pass wholly below the plane, and only the Blue line 
would remain. 

If the turning were continued till half a revolution 
had been accomplished, the Black square would come 
in again. But now it would come up into the plane 
from underneath. It would appear as a Black square 
exactly similar to the first ; but the Orange line, in- 
stead of running left to right from Gold point, would 
run right to left. The square would be the same, only 
differently disposed with regard to the Blue line. It 



112 New Era of Thought. 

would be the looking-glass image of the first square. 
There would be a difference in respect of the lie of the 
particles of which it was composed. If the plane-being 
could examine its thickness, he would find that particles 
which, in the first case, lay above others, now lay below 
them. But, if he were really a plane-being, he would 
have no idea of thickness in his squares, and he would 
find them both quite identical. Only the one would be 
to the other as if it had been pulled through itself. 
In this phenomenon of symmetry he would apprehend 
the difference of the lie of the line, which went in the, 
to him, unknown direction of up-and-down. 



CHAPTER II. 

FURTHER APPEARANCES OF A CUBE TO A 
PLANE- BEING. 

Before leaving the observation of the cube, it is well 
to look at it for a moment as it would appear to a 
plane- being, in whose world there was such a fact 
as attraction. To do this, let the cube rest on the table, 
so that its Dark-blue face is perpendicular in front of 
us. Now, let a sheet of paper be placed in contact with 
the Dark-blue square. Let up and sideways be the 
two dimensions of the plane-being, and away the un- 
known direction. Let the line where the paper meets 
the table, represent the surface of his earth. Then, 
there is to him, as all that he can apprehend of the 
cube, a Dark-blue square standing upright ; and, when 
we look over the edge of the paper, and regard merely 
the part in contact with the paper, we see what the 
plane-being would see. 

If the cube be turned round the up line, the Brown 
line, the Orange line will pass to the near side of 
the paper, and the section made by the cube in the 
paper will be an oblong. Such an oblong can be 
cut out ; and when the cube is fitted into it, it can 
be seen that it is bounded by a Brown line and a 
Blue-green line opposite thereto, while the other boun- 
daries are Black and White lines. Next, if we take 
a section half-way between the Black and White 



114 New Era of Thought. 

squares, we shall have a square cutting the plane of 
the aforesaid paper in a single line. With regard to 
this section, all we have to inquire is, What will take 
the place of this line as the cube turns ? Obviously, the 
line will elongate. From a Dark-blue line it will change 
to a Light-buff line, the colour of the inside of the 
section, and will terminate in a Blue-green point instead 
of a French-grey. Again, it is obvious that, if the cube 
turns round the Orange line, it will give rise to a series 
of oblongs, stretching upwards. This turning can be 
continued till the cube is wholly on the near side of the 
paper, and only the Orange line remains. And, when 
the cube has made half a revolution, the Dark-blue 
square will return into the plane ; but it will run down- 
wards instead of upwards as at first. Thereafter, if 
the cube turn further, a series of oblongs will appear, 
all running downwards from the Orange line. Hence, 
if all the appearances produced by the revolution of the 
cube have to be shown, it must be supposed to be raised 
some distance above the plane-being's earth, so that 
those appearances may be shown which occur when it 
is turned round the Orange line downwards, as well 
as when it is turned upwards. The unknown direction 
comes into the plane either upwards or downwards, but 
there is no special connection between it and either 
of these directions. If it come in upwards, the Brown 
line goes nearwards or — Y ; if it come in downwards, 
or — Z, the Brown line goes away, or Y. 

Let us consider more closely the directions which the 
plane-being would have. Firstly, he would have up-and- 
down, that is, away from his earth and towards it on 
the plane of the paper, the surface of his earth being 
the line where the paper meets the table. Then, if he 
moved along the surface of his earth, there would only 
be a line for him to move in, the line running right and 



Appearances of a Cube to a Plane-Being. 1 1 5 

left. But, being the direction of his movement, he 
would say it ran forwards and backwards. Thus he 
would simply have the words up and down, forwards 
and backwards, and the expressions right and left would 
have no meaning for him. If he were to frame a notion 
of a world in higher dimensions, he must invent new 
words for distinctions not within his experience. 

To repeat the observations already made, let the cube 
be held in front of the observer, and suppose the Dark- 
blue square extended on every side so as to form a 
plane. Then let this plane be considered as independent 
of the Dark-blue square. Now, holding the Brown line 
between finger and thumb, and touching its extremities, 
the Gold and Light-blue points, turn the cube round the 
Brown line. The Dark-blue square will leave the plane, 
the Orange line will tend towards the — Y direction, and 
the Blue line will finally come into the plane pointing 
in the +X direction. If we move the cube so that the 
line which leaves the plane runs -l-Y, then the line 
which befoje ran -t-Y will come into the plane in the 
direction opposite to that of the line which has left the 
plane. The Blue line, which runs in the unknown direc- 
tion can come into either of the two known directions of 
the plane. It can take the place of the Orange line 
by turning the cube round the Brown line, or the place 
of the Brown line by turning it round the Orange line. 
If the plane-being made models to represent these two 
appearances of the cube, he would have identically the 
same line, the Blue line, running in one of his known 
directions in the first model, and in the other of his 
known directions in the second. In studying the cube 
he would find it best to turn it so that the line of un- 
known direction ran in that direction in the positive 
sense. In that case, it would come into the plane in 
the negative sense of . the known directions. 



1 1 6 New Era of Thought. 

Starting with the cube in front of the observer, there 
are two ways in which the Vermilion square can be 
brought into the imaginary plane, that is the extension 
of the Dark-blue square. If the cube turn round the 
Brown line so that the Orange line goes away, {i.e. +Y), 
the Vermilion square comes in on the left of the Brown 
line. If it turn in the opposite direction, the Vermilion 
square comes in on the right of the Brown line. Thus, 
if we identify the plane-being with the Brown line, the 
Vermilion square would appear either behind or before 
him. These two appearances of the Vermilion square 
would seem identical, but they could not be made to 
coincide by any movement in the plane. The diagram 
(Fig. 5.) shows the difference in them. It is obvious that 
no turn in the plane could put one in the place of the 
other, part for part. Thus the plane-being apprehends 
the reversal of the unknown direction by the disposition 
of his figures. If a figure, which lay on one side of a line, 
changed into an identical figure on the other side of it, 
he could be sure that a line of the figure, which at first 
ran in the positive unknown direction, now ran in the 
negative unknown direction. 

We have dwelt at great length on the appearances, 
which a cube would present to a plancbeing, and it will 
be found that all the points which would be likely to 
cause difficulty hereafter, have been explained in this 
obvious case. 

There is, however, one other way, open to a plane- 
being of studying a cube, to which we must attend. 
This is, by steady motion. Let the cube come into the 
imaginary plane, which is the extension of the Dark- 
blue square, i.e. let it touch the piece of paper which 
is standing vertical on the table. Then let it travel 
through this plane at right angles to it at the rate of an 
inch a minute. The plane-being would first perceive 



Appearances of a Cube to a Plane-Being. 117 

a Dark-blue square, that is, he would see the coloured 
lines bounding that square, and enclosed therein would 
be what he would call a Dark-blue solid. In the move- 
ment of the cube, however, this Dark-blue square would 
not last for more than a flash of time. (The edges and 
points on the models are made very large ; in reality 
they must be supposed very minute.) This Dark-blue 
square would be succeeded by one of the colour of the 
cube's interior, i.e. by a Light-buff square. But this 
colour of the interior would not be visible to the plane- 
being. He would go round the square on his plane, and 
would see the bounding lines, viz. Vermilion, White, 
Blue-green, Black. And at the corners he would see 
Deep-yellow, Bright-blue, Crimson, and Blue points. 
These lines and points- would really be those parts of 
the faces and lines of the cube, which were on the point 
of passing through his plane. Now, there would be one 
difference between the Dark-blue square and the Light- 
buff with their respective boundaries. The first only 
lasted for a flash ; the second would last for a minute or 
all but a minute. Consider the Vermilion square. It 
appears to the plane-being as a line. The Brown line 
also appears to him as a line. But there is a difference 
between them. The Brown line only lasts for a flash, 
whereas the Vermilion line lasts for a minute. Hence, 
in this mode of presentation, we may say that for a 
plane-being a lasting line is the mode of apprehending 
a plane, and a lasting plane (which is a plane-being's 
solid) is the mode of apprehending our solids. In the 
same way, the Blue line, as it passes through his plane, 
gives rise to a point. This point lasts for a minute, 
whereas the Gold point only lasted for a flash. 



CHAPTER III. 

FOUR-SPACE, genesis' OF A TESSARACT. ITS REPRE- 
SENTATION IN THREE-SPACE. 

Hitherto we have only looked at Model r. This, with 
the next seven, represent what we can observe of the 
simplest body in Higher Space. A few words will 
explain their construction. A point by its motion traces 
a line. A line by its motion traces either a longer line 
or an area ; if it moves at right angles to its own direc- 
tion, it traces a rectangle. For the sake of simplicity, 
we will suppose all movements to be an inch in length 
and at right angles to each other. Thus, a point moving 
traces a line an inch long ; a line moving traces a square 
inch ; a square moving traces a cubic inch. In these 
cases each of these movements produces something in- 
trinsically different from what we had before. A square 
is not a longer line, nor a cube a larger square. When 
the cube moves, we are unable to see any new direction 
in which it can move, and are compelled to make it 
move in a direction which has previously been used. 
Let us suppose there is an unknown direction at right 
angles to all our known directions, just as a third 
direction would be unknown to a being confined to the 
surface of the table. And let the cube move in this 
unknown direction for an inch. We call the figure it 
traces a Tessaract. The models are representations 
of the appearances a Tessaract would present to us if 
shown in various ways. Consider for a moment what 
happens to a square when moved to form a cube. Each 
of its lines, moved in the new direction, traces a square ; 



Four-Space. Genesis of a Tessaract. 1 1 9 

the square itself traces a new figure, a cube, which ends 
in another square. Now, our cube, moved in a new 
direction, will trace a tessaract, whereof the cube itself 
is the beginning, and another cube the end. These two 
cubes are to the tessaract as the Black square and White 
square are to the cube. A plane-being could not see 
both those squares at once, but he could make models 
of them and let them both rest in his plane at once. So 
also we can make models of the beginning and end of 
the tessaract. Model i is the cube, which is its begin- 
ing ; Model 2 is the cube which is its end. It will be 
noticed that there are no two colours alike in the two 
models. The Silver point corresponds to the Gold point, 
that is, the Silver point is the termination of the line 
traced by the Gold point moving in the new direction- 
The sides correspond in the following manner : — 







Sides. 




Model I. 






Model 2. 


Black corre; 


;ponds to 


Bright-green 


White 






Light-grey 


Vermilion 






Indian-red 


Blue-green 






Yellow-ochre 


Dark-blue 






Burnt-sienna 


Light-yellow 






Dun 



The two cubes should be looked at and compared long 
enough to ensure that the corresponding sides can be 
found quickly. Then there are the following correspon- 
dencies in points and lines. 

Points. 

Model I. Model 2. 

Gold corresponds to Silver 

Fawn „ „ Turquoise 

Terra-cotta „ „ Earthen 

Buff „ „ Blue tint 

Light-blue „ „ Quaker-green 

Dull-purple „ „ Peacock-blue 

Deep-blue „ „ Orange-vermilion 

■RpH .. .. Parole 



I20 



New Era of Thought. 



Lines. 



Model I. 






Orange 


corresponds to 


Crimson 


» )j 


Green-grey 


J) 


» 


Blue 


)> 


) 


Brown 


)} 


) 


French -grey 


j> 


> 


Dark-slate 


>j 


» 


Green 


») 


> 


Reddish 


)) 


J 


Bright-blue 


» ) 




Leaden 


» 


» 


Deep-yellow 


j> ) 


» 



Model 2. 
Leaf-green 
Dull-green 
Dark-purple 
Purple-brown 
Dull-blue 
Dark-pink 
Pale-pink 
Indigo 
Brown-green 
Dark-green 
Pale-yellow 
Dark 



The colour of the cube itself is invisible, as it is 
covered by its boundaries. We suppose it to be Sage- 
green. 

These two cubes are just as disconnected when looked 
at by us as the black and white squares would be to a 
plane-being if placed side by side on his plane. He 
cannot see the squares in their right position with regard 
to each other, nor can we see the cubes in theirs. 

Let us now consider the vermilion side of Model i. 
If it move in the X direction, it traces the cube of 
Model I. Its Gold point travels along the Orange line, 
and itself, after tracing the cube, ends in the Blue-green 
square. But if it moves in the new direction, it will 
also trace a cube, for the new direction is at right angles 
to the up and away directions, in which the Brown and 
Blue lines run. Let this square, then, move in the 
unknown direction, and trace a cube. This cube we 
cannot see, because the unknown direction runs out of 
our space at once, just as the up direction runs out of 
the plane of the table. But a plane-being could see the 
square, which the Blue line traces when moved upwards, 
by the cube being turned round the Blue line, the 



Four-Space. Genesis of a Tessaract. 121 

Orange line going upwards ; then the Brown line comes 
into the plane of the table in the - X direction. So 
also with our cube. As treated above, it runs from the 
Vermilion square out of our space. But if the tessaract 
were turned so that the line which runs from the Gold 
point in the unknown direction lay in our space, and the 
Orange line lay in the unknown direction, we could then 
see the cube which is formed by the movement of the 
Vermilion square in the new direction. 

Take Model 5. There is on it a Vermilion square. 
Place this so that it touches the Vermilion square on 
Model r. All the marks of the two squares are 
identical. This Cube 5, is the one traced by the 
Vermilion square moving in the unknown direction. In 
Model 5, the whole figure, the tessaract, produced by 
the movement of the cube in the unknown direction, is 
supposed to be so turned that the Orange line passes 
into the unknown direction, and that the line which 
went in the unknown direction, runs opposite to the old 
direction of the Orange line. Looking at this new cube, 
we see that there is a Stone line running to the left from 
the Gold point. This line is that which the Gold point 
traces when moving in the unknown direction. 

It is obvious that, if the Tessaract turns so as to show 
us the side, of which Cube 5 is a model, then Cube i will 
no longer be visible. The Orange line will run in the 
unknown or fourth direction, and be out of our sight, 
together with the whole cube which the Vermilion 
square generates, when the Gold point moves along the 
Orange line. Hence, if we consider these models as real 
portions of the tessaract, we must not have more thap 
one before us at once. When we look at one, the others 
must necessarily be beyond our sight and touch. But 
we may consider them simply as models, and, as such, 
we may let them lie alongside of each other. In this 



122 



New Era of Thought. 



case, we must remember that their real relationships are 
not those in which we see them. 

We now enumerate the sides of the new Cube 5, so 
that, when we refer to it, any colour may be recognised 
by name. 

The square Vermilion traces a Pale-green cube, and 
ends in an Indian-red square. 

(The colour Pale-green of this cube is not seen, as it 
is entirely surrounded by squares and lines of colour.) 

Each Line traces a Square and ends in a Line. 



The Blue line\ 
„ Brown „ » 
„ Deep-yellow,, g 
„ Green „ 


Light-brown square 
Yellow 
■ Light-red „ 
Deep-crimson „ 


0) 


Purple-brown line 
Dull-blue „ 
Dark „ 
.Indigo „ 


Each Point traces a Line and ends in a Point. 


The Gold point 
„ Buff 

„ Light-blue „ 
„ Red 


C4 

i 


Stone line 
Light-green „ 
Rich-red „ 
Emerald „ 


.S 

S' 


Silver point 
Blue-tint „ 
Quaker-green „ 
Purple ■ „ 



It will be noticed that besides the Vermilion square of 
this cube another square of it has been seen before. A 
moment's comparison with the experience of a plane- 
being will make this more clear. If a plane-being has 
before him models of the Black and White squares of the 
Cube, he sees that all the colours of the one are different 
from all the colours of the other. Next, if he looks at 
a model of the Vermilion square, he sees that it starts 
from the Blue line and ends in a line of the White square, 
the Deep-yellow line. In this square he has two lines 
which he had before, the Blue line with Gold and Buff 
points, the Deep-yellow line with Light-blue and Red 
points. To him the Black and White squares are his 
Models I and 2, and the Vermilion square is to him as 
our Model 5 is to us. The left-hand square of Model S 
is Indian-red, and is identical with that of the same 



Four-Space. Genesis of a Tessaract, 123 

colour on the left-hand side of Model 2. In fact, Model 
5 shows us what lies between the Vermilion face of i, 
and the Indian-red face of 2. 

From the Gold point we suppose four perfectly in- 
dependent lines to spring forth, each of them at right 
angles to all the others. In our space there is only- 
room for three lines mutually at right angles. It will 
be found, if we try to introduce a fourth at right angles 
to each of three, that we fail ; hence, of these four 
lines one must go out of the space we know. The 
colours of these four lines are Brown, Orange, Blue, 
Stone. In Model i are shown the Brown, Orange, and 
Blue. In Model S are shown the Brown, Blue, and 
Stone. These lines might have had any directions at 
first, but we chose to begin with the Brown line going 
up, or Z, the Orange going X, the Blue going Y, and the 
Stone line going in the unknown direction, which we 
will call W. 

Consider for a moment the Stone and the Orange' 
lines. They can be seen together on Model 7 by look- 
ing at the lower face of it. They are at right angles to 
each other, and if the Orange line be turned to take 
the place of the Stone line, the latter will run into the 
negative part of the direction previously occupied by 
the former. This is the reason that the Models 3, 5, 
and 7 are made with the Stone line always running in 
the reverse direction of that line of Model i, which is 
wanting in each respectively. It will now be easy to 
find out Models 3 and 7. All that has to be done is, to 
discover what faces they have in common with i and 2, 
and these faces will show from which planes of i they 
are generated by motion in the unknown direction. 

Take Model 7. On one side of it there is a Dark- 
blue square, which is identical with the Dark-blue 
square of Model i. Placing it so that it coincides with 



124 New Era of Thought. 

I by this square line for line, we see that the square 
nearest to us is Burnt-sienna, the same as the near 
square on Model 2. Hence this cube is a model of 
what the Dark-blue square traces on moving in the 
unknown direction. Here the unknown direction co- 
incides with the negative away direction. In fact, to 
see this cube, we havelbeen obliged to suppose the Blue 
line turned into the unknown direction, for we cannot 
look at more than three of these rectangular lines at 
once in our space, and in this Model 7 we have the 
Brown, Orange, and Stone lines. The faces, lines, and 
points of Cube 7 can be identified by the following list. 

The Dark-blue square traces a Dark-stone cube 
(whose interior is rendered invisible by the bounding 
squares), and ends in a Burnt-sienna square. 

Each Line traces a Square and ends in a Line. 



The Orange line'\ 
„ Brown 
„ French-grey , 
,, Reddish 



Azure square 

Yellow „ 

Yellow-green „ 
l,Ochre „ 



Leaf-green line 
Dull-blue „ 
Dark-pink „ 
, Brown-green „ 



Each Point traces a Line and ends in a Point. 



The Gold poinfj f Stone line 

„ Fawn „ I g I Smoke „ 

„ Light-blue „ j | 1 Rich-red „ 

„ Dull-purple „ j I Green-blue „ 



Silver point 

Turquoise „ 

Quaker-green „ 
Peacock-blue „ 



If we now take Model 3, we see that it has a Black 
square uppermost, and has Blue and Orange lines. 
Hence, it evidently proceeds from the Black square in 
Model I ; and it has in it Blue and Orange lines, which 
proceed from the Gold point. But besides these, it has 
running downwards a Stone line. The line wanting is 
the Brown line, and, as in the other cases, when one of 
the three lines of Model i turns out into the unknown 
direction, the Stone line turns into the direction op- 
posite to that from which the line has turned. Take 



Four-Space. Genesis of a Tessarad. 125 

this Model 3 and place it underneath Model i, raising 
the latter so that the Black squares on the two coincide 
line for line. Then we see what would come into our 
view if the Bfown line were to turn into the unknown 
direction, and the Stone line come into our space down- 
wards. Looking at this cubcj we see that the following 
parts of the tessaract have been generated. 

The Black square traces a Brick-red cube (invisible 
because covered by its own sides and edges), and ends 
in a Bright-green square. 

Each Line traces a Square and ends in a Line. 

■ Leaf-green line 

Dull-green „ 

Dark-purple „ 

i Purple-brown „ 



The Orange line'\ 
„ Crimson , 
„ Green-grey , 
,, Blue , 



Azure square\ 
Rose ,. 

Seai-blue ,, 

Light-brown ,, 



Each Point traces a Line and ends in a Point. 



The Gold point' 

„ Fawn 
„ Terra-cotta 
„ Buff 



I Stone line'\ ^ ( Silver point 

Smoke „ 1^1 Tut'quoise „ 

Magenta „ | S 1 Earthen „ 

Light-green „ j | (.Blue^tint „ 

This completes the enumeration of the regions of 
Cube 3. It may seem a little unnatural that it should 
come in downwards ; but it must be remembered that 
the new fourth direction has no more relation to up-and- 
down than to right-and-left or to near-and-far. 

And if, instead of thinking of a plane-being as living 
on the surface of a table, we suppose his world to be the 
surface of the sheet of paper touching the Dark-blue 
square of Cube i, then we see that a turn round the 
Orange line, which makes the Brown line go into the 
plane-being's unknown direction, brings the Blue line 
into his downwards direction. 

There still remain to be described Models 4, 6, and 8. 
It will be shown that Model 4 is to Model 3 what 
Model 2 is to Model i. That is, if, when 3 is In our 



126 New Era of Thought. 

space, it be moved so as to trace a tessaract, 4 will be 
the opposite cube in which the tessaract ends. There 
is no colour common to 3 and 4. Similarly, 6 is the 
opposite boundary of the tessaract generated by S, and 
8 of that by 7. 

A little closer consideration will reveal several points. 
Looking at Cube S, we see proceeding from the Gold 
point a Brown, a Blue, and a Stone line. The Orange 
line is wanting ; therefore, it goes in the unknown 
direction. If we want to discover what exists in the 
unknown direction from Cube 5, we can get help from 
Cube I. For, since the Orange line lies in the unknown 
direction from Cube 5, the Gold point will, if moved 
along the Orange line, pass in the unknown direction. 
So also, the Vermilion square, if moved along in the 
direction of the Orange line, will proceed in the un- 
known direction. Looking at Cube i we see that 
the Vermilion square thus moved ends in a Blue-green 
square. Then, looking at Model 6, on it, corresponding 
to the VermiHon square on Cube S, is a Blue-green 
square. 

Cube 6 thus shows what exists an inch beyond S 
in the unknown direction. Between the right-hand 
face on 5 and the right-hand face on 6 lies a cube, the 
one which is shown in Model i. Model i is traced by 
the Vermilion square moving a;n inch along the direc- 
tion of the Orange line. In Model S, the Orange line 
goes in the unknown direction ; and looking at Model 6 
we see what w-e should get at the end of a jhovement of 
one inch in that direction. We have still to enumerate 
the colours of Cubes 4, 6, and 8, and we do so in the 
following list. In the first column is designated the 
part of the cube ; in the columns under 4, 6, 8, come the 
colours which 4, 6, 8, respectively have in the parts 
designated in the corresponding line in the first column. 



Four-Space. Genesis of a Tessaract. 127 



Cube itself: — 

Squares : — 
Lower face 
Upper 
Left-hand 
Right-hand 
Near 
Far 



Chocolate 



Light-grey 

White 

Light-red 

Deep-brown 

Ochre 

Deep-green 

Lines : — 

On ground, going round 
right : — 

4 6 

1. Brown-green Smoke 

2. Dark-green Crimson 

3. Pale-yellow Magenta 

4. Dark Dull-green 



Oak-yellow 

Rose 

Deep-brown 

Yellow-ochre 

Blue-green 

Yellow-green 

Dark-grey 



Salmon 

Sea-blue 

Deep-green 

Deep-crimson 

Dark-grey 

Dun 

Light-yellow 



the square from left to 



Dark-purple 
Magenta 
Green-grey 
Light-green 



Vertical, going round the sides from left to right : — 

1. Rich-red Dark-pink Indigo 

2. Green-blue French-grey Pale-pink 

3. Sea-green Dark-slate Dark-slate 

4. Emerald Pale-pink Green 

Round upper face in same order : — 

1. Reddish Green-blue Pale-yellow 

2. Bright-blue Bright-blue Sea-green 

3. Leaden Sea-green Leaden 

4. Deep-yellow Dark-green Emerald 

Points : — 

On lower face, going from left to right 



1. Quaker-green 

2. Peacock-blue 

3. Orange-vermilion 

4. Purple 

On upper face : — 

1. Light-blue 

2. Dull-purple 

3. Deep-blue 

4. Red 



Turquoise 
Fawn 

Terra-cotta 
Earthen 



Peacock-blue 
Dull-purple 
Deep-blue 
Orange-vermilion 



Blue-tint 
Earthen 
Terra-cotta 
Buff 



Purple 

Orange-vermilion 
Deep-blue 
Red 



128 New Era of Thought. 

If any one of these cubes be taken at random, it is 
easy enough to find out to what part of the Tessaract 
it belongs. In all of them, except 2, there will be one 
face, which is a copy of a face on i ; this face is, in 
fact, identical with the face on i which it resembles. 
And the model shows what lies in the unknown 
direction from that face. This unknown direction is 
turned into our space, so that we can see and touch the 
result of moving a square in it. And we have sacrificed 
one of the three original directions in order to do this. 
It will be found that the line, which in i goes in the 
4th direction, in the other models always runs in a 
negative direction. 

Let us take Model 8, for instance. Searching it for 
a face we know, we come to a Light-yellow face away 
from us. We place this face parallel with the Light- 
yellow face on Cube i, and we see that it has a Green 
line going up, and a Green-grey line going to the right 
from the Buff point. In these respects it is identical 
with the Light-yellow face on Cube i. But instead of 
a Blue line coming towards us from the Buff point, 
there is a Light-green line. This Light-green line, then, 
is that which proceeds in the unknown direction from 
the Buff point. The line is turned towards us in this 
Model 8 in the negative Y direction ; and looking at 
the model, we see exactly what is formed when in the 
motion of the whole cube in the unknown direction, 
the Light-yellow face is moved an inch in that direction. 
It traces out a Salmon cube {v. Table on p. 127), and it 
has Sea-blue and Deep-green sides below and above, 
and Deep-crimson and Dark-grey sides left and right, 
and Dun and Light-yellow sides near and far. If we 
want to verify the correctness of any of these details, we 
must turn to Models i and 2. What lies an inch from 
the Light-yellow square in the unknown direction .' 



Four-Space. Genesis of a Tessaract. 1 29 

Model 2 tells us, a Dun square. Now, looking at 8, 
we see that towards us lies a Dun square. This is what 
lies an inch in the unknown direction from the Light- 
yellow square. It is here turned to face us, and we 
can see what lies between it and the Light-yellow 
square. 



CHAPTER IV. 

TESSARACT MOVING THROUGH THREE-SPACE. 
MODELS OF THE SECTIONS. 

In order to obtain a clear conception of the higher 
solid, a certain amount of familiarity with the facts 
shown in these models is necessary. But the best way 
of obtaining a systematic knowledge is shown here- 
after. What these models enable us to do, is to take a 
general review of the subject. In all of them we see 
simply the boundaries of the tessaract in our space ; 
we can no more see or touch the tessaract's solidity 
than a plane-being can touch the cube's solidity. 

There remain the four models 9, 10, 11, 12. Model 9 
represents what lies between i and 2. If i be moved 
an inch in the unknown direction, it traces out the 
tessaract and ends in 2. But, obviously, between i and 
2 there must be an infinite number of exactly similar 
solid sections ; these are all like Model 9. 

Take the case of a plane-being on the table. He 
sees the Black square, — that is, he sees the lines round 
it, — and he knows that, if it moves an inch in some 
mysterious direction, it traces a new kind of figure, the 
opposite boundary whereof is the White square. If, 
then, he has models of the White and Black squares, 
he has before him the end and beginning of our cube. 
But between these squares are any number of others, 
the plane sections of the cube. We can see what they 



Tessaract Moving in Three-Space. 131 

are. The interior of each is a Light-buff (the colour 
of the substance of the cube), the sides are of the colours 
of the vertical faces of the cube, and the points of the 
colours of the vertical lines of the cube, viz.. Dark-blue, 
Blue-green, Light-yellow, Vermilion lines, and Brown, 
French-grey, Dark-slate, Green points. Thus, the square, 
in moving in the unknown direction, traces out a 
succession of squares, the assemblage of which makes 
the cube in layers. So also the cube, moving in the 
unknown direction, will at any point of its motion, 
still be a cube ; and the assemblage of cubes thus placed 
constitutes the tessaract in layers. We suppose the 
cube to change its colour directly it begins to move. 
Its colour between i and 2 we can easily determine 
by finding what colours its different parts assume, as 
they move in the unknown direction. The Gold point 
immediately begins to trace a Stone-line. We will 
look at Cube 5 to see what the Vermilion face becomes ; 
we know the interior of that cube is Pale-green {^. Table, 
p. 122). Hence, as it moves in the unknown direction, 
the Vermilion square forms in its course a series of 
Pale-green squares. The Brown line gives rise to a 
Yellow square ; hence, at every point of its course in 
the fourth direction, it is a Yellow line, until, on taking 
its final position, it becomes a Dull-blue line. Looking 
at Cube 5, we see that the Deep yellow line becomes 
a Light-red line, the Green line a Deep Crimson one, 
the Gold point a Stone one, the Light-blue point a 
Rich-red one, the Red point an Emerald one, and the 
Buff point a Light-green one. Now, take the Model 9. 
Looking at the left side of it, we see exactly that into 
which the Vermilion square is transformed, as it moves 
in the unknown direction. The left side is an exact 
copy of a section of Cube 5, parallel to the Vermilion 
face. 



132 A New Era of Thought. 

But we have only accounted for one side of our 
Model 9. There are five other sides. Take the near 
side corresponding to the Dark- blue square on Cube i. 
When the Dark-blue square moves, it traces a Dark- 
stone cube, of which we have a copy in Cube 7. Look- 
ing at 7 {v. Table, p. 124), we see that, as soon as the 
Dark-blue square begins to move, it becomes of a Dark- 
stone colour, and has Yellow, Ochre, Yellow-green, and 
Azure sides, and Stone, Rich-red, Green-blue, Smoke 
lines running in the unknown direction from it. Now, 
the side of Model 9, which faces us, has these colours 
the squares being seen as lines, and the lines as points. 
Hence Model 9 is a copy of what the cube becomes, 
so far as the Vermilion and Dark-blue sides are con- 
cerned, when, moving in the unknown direction, it 
traces the tessaract. 

We will now look at the lower square of our model. 
It is a Brick-red square, with Azure, Rose, Sea-blue, 
and Light-brown lines, and with Stone, Smoke, Magenta, 
and Light-green points. This, then, is what the Black 
square should change into, as it moves in the unknown 
direction. Let us look at Model 3. Here the Stone 
line, which is the line in the unknown direction, runs 
downwards. It is turned into the downwards direction, 
so that the cube traced by the Black square may be 
in our space. The colour of this cube is Brick-red ; 
the Orange line has traced an Azure, the Blue line a 
Light-brown, the Crimson line a Rose, and the Green- 
grey line a Sea-blue square. Hence, the lower square 
of Model 9 shows what the Black square becomes, as 
it traces the tessaract ; or, in other words, the section 
of Model 3 between the Black and Bright-green squares 
exactly corresponds to the lower face of Model 9, 

Therefore, it appears that Model 9 is a model of a 
section of the tessaract, that it is to the tessaract what 



Tessaract Moving in Three-Space. 133 

a square between the Black and White squares is to 
the cube. 

To prove the other sides correct, we have to see what 
the White, Blue-green, and Light-yellow squares of 
Cube I become, as the cube moves in the unknown 
direction. This can be effected by means of the Models 
4, 6, 8. Each cube can be used as an index for showing 
the changes through which any side of the first model 
passes, as it moves in the unknown direction till it 
becomes Cube 2. Thus, what becomes of the White 
square .' Look at Cube 4. From the Light-blue corner 
of its White square runs downwards the Rich-red line 
in the unknown direction. If we take a parallel section 
below the White square, we have a square bounded 
by Ochre, Deep-brown, Deep-green, and Light-red 
lines ; and by Rich-red, Green-blue, Sea-green, and 
Emerald points. The colour of the cube is Chocolate, 
and therefore its section is Chocolate. This description 
is exactly true of the upper surface of Model 9. 

There still remain two sides, those corresponding to 
the Light-yellow and Blue-green of Cube i. What the 
Blue-green square becomes midway between Cubes i and 
2 can be seen on Model 6. The colour of the last-named 
is Oak-yellow, and a section parallel to its Blue-green 
side is surrounded by Yellow-green, Deep-brown, Dark- 
grey and Rose lines and by Green-blue, Smoke, Magenta, 
and Sea-green points. This is exactly similar to the 
right side of Model 9. Lastly, that which becomes of 
the Light-yellow side can be seen on Model 8. The sec- 
tion of the cube is a Salmon square bounded by Deep- 
crimson, Deep-green, Dark-grey and Sea-blue lines and 
by Emerald, Sea-green, Magenta, and Light-green points. 

Thus the models can be used to answer any question 
about sections. For we have simply to take, instead of 
the whole cube, a plane, and the relation of the whole 



134 A New Era of Thought. 

tessaract to that plane can be told by looking at the 
model, which, starting with that plane, stretches from it 
in the unknown direction. 

We have not as yet settled the colour of the interior 
of Model 9. It is that part of the tessaract which is 
traced out by the interior of Cube i. The unknown 
direction starts equally and simultaneously from every 
point of every part of Cube i, just as the up direction 
starts equally and simultaneously from every point of a 
square. Let us suppose that the cube, which is Light- 
buff, changes to a Wood-colour directly it begins to trace 
the tessaract Then the internal part of the section be- 
tween I and 2 will be a Wood-colour. The sides of the 
Model 9 are of the greatest importance. They are the 
colour of the six cubes, 3, 4, 5, 6, 7, and 8. The colours 
of I and 2 are wanting, viz. Light-buff and Sage-green. 
Thus the section between i and 2 can be found by its 
wanting the colours of the Cubes i and 2. 

Looking at Models 10, 11, and 12 in a similar manner, 
the reader will find they represent the sections between 
Cubes 3 and 4, Cubes 5 and 6, and Cubes 7 and 8 re- 
spectively. 



CHAPTER V. 

REPRESENTATION OF THREE-SPACE BY NAMES, AND 
IN A PLANE. 

We may now ask ourselves the best way of passing on 
to a clear comprehension of the facts of higher space. 
Something can be effected by looking at these models ; 
but it is improbable that more than a slight sense of 
analogy will be obtained thus. Indeed, we have been 
trusting hitherto to a method which has something 
vicious about it — we have been trusting to our sense of 
what must be. The plan adopted, as the serious effort 
towards the comprehension of this subject, is to learn a 
small portion of higher space. If any reader feel a diffi- 
culty in the foregoing chapters, or if the subject is to be 
taught to young minds, it is far better to abandon all 
attempt to see what higher space must be, and to learn 
what it is from the following chapters. 

Naming a Piece of Space. 

The diagram (Fig. 6) represents a block of 27 cubes, 
which form Set i of the 81 cubes. The cubes are 
coloured, and it will be seen that the colours are ar- 
ranged after the pattern of Model i of previous chapters, 
which will serve as a key to the block. In the diagram, G. 
denotes Gold, O. Orange, F. Fawn, Br. Brown, and so on. 
We will give names to the cubes of this block. They 



'36 



A New Era of Thought. 



should not be learnt, but kept for reference. We will 
write these names in three sets, the lowest consisting of 
the cubes which touch the table, the next of those im- 
mediately above them, and the third of those at the top. 
Thus the Gold cube is called Corvus, the Orange, Cuspis, 
the Fawn, Nugae, and the central one below, Syce. The 
corresponding colours of the following set can easily be 
traced. 



Olus 


Semita 


Lama 


Via 


Mel 


Iter 


Ilex 


Callis 


Sors 


Bucina 


Murex 


Daps 


Alvus 


Mala 


Proes 


Arctos 


Moena 


Far 


Cista 


Cadus 


Crus 


Dos 


Syce 


Bolus 


Corvus 


Cuspis 


NugEe 



Thus the central or Light-buff cube is called Mala; the 
middle one of the lower face is Syce ; of the upper face 
Mel ; of the right face, Proes ; of the left, Alvus ; of the 
front, Moena (the Dark-blue square of Model i) ; and of 
the back, Murex (the Light-yellow square). 

Now, if Model i be taken, and considered as represent- 
ing a block of 64 cubes, the Gold corner as one cube, the 
Orange line as two cubes, the Fawn point as one cube^ 
the Dark-blue square as four cubes, the Light-buff interior 
as eight cubes, and so on, it will correspond to the dia- 
gram (Fig. 7). This block differs from the last in the 
number of cubes, but the arrangement of the colours is 
the same. The following table gives the names which 
we will use for these cubes. There are no new names ; 
they are only applied more than once to all cubes of the 
same colour. 







/ 


-■ / y y 


/ 






/ 


/ / 1 


J 








/ / / 1 


J 


r / / 


J 

1 
) 

1 
1 


/ 


/ / / 1 




/ / / 


LBl. 


R 


1 


, 


/ / / A 


K 


D.P. 




R. 


... / 


L.Bl. 


■ Br 


O.Bl. 


0.81 


F.G.. 


i 


Br 


O.Bl. 


r.c, 




Bt. 


O.Bl. 


D.Bl. 


f:gt 

F. 


: 


G 





' / 


1 


G 
























J 



F,c 6 



F,^ 7. 



/ 


/ / / > 


' / / 


/ / / / 


/ / 


^ / / / 


/ / 1 


/ / / / / /WA 


/ / / / 


/ / / 


LBl 


R 


R 


R 


O.F. / / 


Bt. 


DBl. 


O.BI 


D.Bl. 


F.Gr. / /' 

' 1 


Br. 


D.Bl. 


D.B1. 


D.B.I. 


K,./; 


Br. 


DBl. 


D.BI, 


D.Bl 


re,.// 


C 











'I 



Iff. 8. 



^'S 



[To/acep. 136. 



Three-Space by Names, and in a Plane. 137 



i 


Olus 


Semita 


Semita 


Lama 


Fourth 


)Via 


Mel 


Mel 


Iter 


Floor. ' 


Wra 


Mel 


Mel 


Iter 


1 


Ulex 


Callis 


Callis 


Sors 




r Bucina 


Murex 


Murex 


Daps 


Third ' 


\ Alvus 


Mala 


Mala 


Proes 


Floor. 


\ Alvus 


Mala 


Mala 


Proes 




' Arctos 


Moena 


Moena 


Far 




! Bucina 


Murex 


Murex 


Daps 


Second 


\ Alvus 


Mala' 


Mala 


Proes 


Floor. 


S Alvus 


Mala 


Mala 


Proes 




' Arctos 


Moena 


Mcena 


Far 


1 


r Cista 


Cadus 


Cadus 


Crus 


First 


\dos 


Syce 


Syce 


Bolus 


Floor. 


y Dos 


Syce 


Syce 


Bolus 




' Corvus 


Cuspis 


Cuspis 


Nugse 



If we now consider Model i to represent a block, five 
cubes each way, built up of inch cubes, and colour it in 
the same way, that is, with similar colours for the corner- 
cubes, edge-cubes, face-cubes, and interior-cubes, we 
obtain what is represented in the diagram (Fig. 8). 
Here we have nine Dark-blue cubes called Moena ; that 
is, Moena denotes the nine Dark-blue cubes, forming a 
layer on the front of the cube, and filling up the whole 
front except the edges and points. Cuspis denotes three 
Orange, Dos three Blue, and Arctos three Brown cubes. 

Now, the block of cubes can be similarly increased to 
any size we please. The corners will always consist of 
single cubes ; that is, Corvus will remain a single cubic 
inch, even though the block be a hundred inches each 
way. Cuspis, in that case, will be 98 inches long, and 
consist of a row of 98 cubes ; Arctos, also, will be a long 
thin line of cubes standing up. Moena will be a thin 
layer of cubes almost covering the whole front of the 
block ; the number of them will be 98 times 98. Syce 



138 A New Era of Thought. 

will be a similar square layer of cubes on the ground, so 
also Mel, Alvus, Proes, and Murex in their respective 
places. Mala, the interior of the cube, will consist of 
98 times 98 times 98 inch cubes. 

Now, if we continued in this manner till we had a 
very large block of thousands of cubes in each side 
Corvus would, in comparison to the whole block, be a 
minute point of a cubic shape, and Cuspis would be a 
mere line of minute cubes, which would have length, but 
very small depth or height. Next, if we suppose this 
much sub-divided block to be reduced in size till it be- 
comes one measuring an inch each way, the cubes of 
which it consists must each of them become extremely 
minute, and the corner cubes and line cubes would be 
scarcely discernible. But the cubes on the faces would 
be just as visible as before. For instance, the cubes com- 
posing Moena would stretch out on the face of the cube 
so as to fill it up. They would form a layer of extreme 
thinness, but would cover the face of the cube (all of it 
except the minute lines and points). Thus we may use 
the words Corvus and Nugae, etc., to denote the corner- 
points of the cube, the words Mcena, Syce, Mel, Alvus, 
Proes, Murex, to denote the faces. It must be remem- 
bered that these faces have a thickness, but it is ex- 
tremely minute compared with the cube. Mala would 
denote all the cubes of the interior except those, which 
compose the faces, edges, and points. Thus, Mala would 
practically mean the whole cube except the colouring on 
it. And it is in this sense that these words will be used. 
In the models, the Gold point is intended to be a Corvus, 
only it is made large to be visible ; so too the Orange 
line is meant for Cuspis, but magnified for the same 
reason. Finally, the 27 names of cubes, with which we 
began, come to be the names of the points, lines, and 
faces of a cube, as shown in the diagram (Fig. 9). With 




[Tofacep. 13S. 



Three-Space by Names, and in a Plane. 139 

these names it is easy to express what a plane-being 
would see of any cube. Let us suppose that McEna is, 
only of the thickness of his matter. We suppose his 
matter to be composed of particles, which slip about on 
his plane, and are so thin that he cannot by any means 
discern any thickness in them. So he has no idea of 
thickness. But we know that his matter must have some 
thickness, and we suppose Moena to be of that degree of 
thickness. If the cube be placed so that Mcena is in his 
plane, Corvus, Cuspis, Nugae, Far, Sors, Callis, Ilex and 
Arctos will just come into his apprehension; they will be 
like bits of his matter, while all that is beyond them in 
the direction he does not know, will be hidden from him. 
Thus a plane-being can only perceive the Mcena or Syce 
or some one other face of a cube ; that is, he would take 
the Moena of a cube to be a solid in his plane-space, and 
he would see the lines Cuspis, Far, Callis, Arctos. To him 
they would bound it. The points Corvus, Nugse, Sors, 
and Ilex, he would not see, for they are only as long as 
the thickness of his matter, and that is so slight as to be 
indiscernible to him. 

We must now go with great care through the exact 
processes by which a plane-being would study a cube. 
For this purpose we use square slabs which have a cer- 
tain thickness, but are supposed to be as thin as a plane- 
being's matter. Now, let us take the first set of 8 1 cubes 
again, and build them from i to 27. We must realize 
clearly that two kinds of blocks can be built. It may 
be built of 27 cubes, each similar to Model i, in which 
case each cube has its regions coloured, but all the cubes 
are alike. Or it may be built of 27 differently coloured 
cubes like Set i, in which case each cube is coloured 
wholly with one colour in all its regions. If the latter 
set be used, we can still use the names Mcena, Alvus, etc. 
to denote the front, side, etc., of any one of the cubes. 



140 



A New Era of Thought. 



whatever be its colour. When they are built up, place 
a piece of card against the front to represent the plane 
on which the plane-being lives. The front of each of 
the cubes in the front of the block touches the plane. 
In previous chapters we have supposed Moena to be a 
Blue square. But we can apply the name to the front 
of a cube of any colour. Let us say the Moena of each 
front cube is in the plane ; the Mcena of the Gold cube 
is Gold, and so on. To represent this, take nine slabs 
of the same colours as the cubes. Place a stiff piece of 
cardboard (or a book-cover) slanting from you, and put 
the slabs on it. They can be supported on the incline 
so as to prevent their slipping down away from you by 
a thin book, or another sheet of cardboard, which stands 
for the surface of the plane-being's earth. 

We will now give names to the cubes of Block i of 
the 81 Set. We call each one Mala, to denote that it is 
a cube. They are written in the following list in floors 
or layers, and are supposed to run backwards or away 
from the reader. Thus, in the first layer, Frenuni Mala 
is behind or farther away than Urna Mala ; in the 
second layer, Ostrum is in front. Uncus behind it, and 
Ala behind Uncus. 



Third, or ( Mars Mala 
Top < Spicula Mala 
Floor. (. Comes Mala 



Merces Mala 
Mora Mala 
Tibicen Mala 



Tyro Mala 
Oliva Mala 
Vestis Mala 



Second, or ( Ala Mala 
.Middle < Uncus Mala 
Floor. ' Ostrum Mala 



Cortis Mala 
Pallor Mala 
Bidens Mala 



Aer Mala 
Tergum Mala 
Scena Mala 



First, or ( Sector Mala 
Bottom < Frenum Mala 
Floor. ( Urna Mala 



Hama Mala 
Plebs Mala 
Moles Mala 



Remus Mala 
Sypho Mala 
Saltus Mala 



These names should be learnt so that the different 
cubes in the block can be referred to quite easily and 



-z 



-X- 



Urna 



MoUi 



Ostrunt 



FglO. 



\_Toface p. 141. 



Three-Space by Names, and in a Plane. 141 

immediately by name. They must be learnt in every 
order, that is, in each of the three directions backwards 
and forwards, e.g. Urna to Saltus, Urna to Sector, Urna 
to Comes ; and the same reversed, viz., Comes to Urna, 
Sector to Urna, etc. Only by so learning them can 
the mind identify any one individually without even a 
momentary reference to the others around it. It is well 
to make it a rule not to proceed from one cube to a 
distant one without naming the intermediate cubes. 
For, in Space we cannot pass from one part to another 
without going through the intermediate portions. And, 
in thinking of Space, it is well to accustom our minds to 
the same limitations. 

Urna Mala is supposed to be solid Gold an inch each 
way ; so too all the cubes are supposed to be entirely of 
the colour which they show on their faces. Thus any 
section of Moles Mala will be Orange, of Plebs Mala 
Black, and so on. 

Let us now draw a pair of lines on a piece of paper 
or cardboard like those in the diagram (Fig. 10). In 
this diagram the top of the page is supposed to rest on 
the table, and the bottom of the page to be raised and 
brought near the eye, so that the plane of the diagram 
slopes upwards to the reader. Let Z denote the upward 
•direction, and X the direction from left to right. Let 
us turn the Block of cubes with its front upon this 
slope i.e. so that Urna fits upon the square marked 
Urna. Moles will be to the right and Ostrum above 
Urna, i.e. nearer the eye. We might leave the block as it 
stands and put the piece of cardboard against it ; in this 
case our plane-world would be vertical. It is difficult to 
fix the cubes in this position on the plane, and therefore 
more convenient if the cardboard be so inclined that 
they will not slip off. But the upward direction must 
be identified with Z. Now, taking the slabs, let us 



142 A New Era of Thought. 

compose what a plane-being would see of the Block. 
He would perceive just the front faces of the cubes of 
the Block, as it comes into his plane ; these front faces 
we may call the Moenas of the cubes. Let each of the 
slabs represent the Moena of its corresponding cube, the 
Gold slab of the Gold cube and so on. They are thicker 
than they should be ; but we must overlook this and 
suppose we simply see the thickness as a line. We thus 
build a square of nine slabs to represent the appearance 
to a plane-being of the front face of the Block. The 
middle one, Bidens Moena, would be completely hidden 
from him by the others on all its sides, and he would 
see the edges of the eight outer squares. If the Block 
now begin to move through the plane, that is, to cut 
through the piece of paper at right angles to it, it will 
not for some time appear any different. For the sections 
of Urna are all Gold like the front face Moena, so that 
the appearance of Urna at any point in its passage will 
be a Gold square exactly like Urna Moena, seen by the 
plane-being as a line. Thus, if the speed of the Block's 
passage be one inch a minute, the plane-being will see 
no change for a minute. In other words, this set of 
slabs lasting one minute will represent what he sees. 

When the Block has passed one inch, a different set 
of cubes appears. Remove the front layer of cubes. 
There will now be in contact with the paper nine new 
cubes, whose names we write in the order in which we 
should see them through a piece of glass standing up- 
right in front of the Block : 

Spicula Mala Mora Mala Oliva Mala 

Uncus Mala Pallor Mala Tergum Mala 

Frenum Mala Plebs Mala Sypho Mala 

We pick out nine slabs to represent the Moenas of 
these cubes, and placed in order they show what the 



Three-Space by Names, and in a Plane. 143 

plane-being sees of the second set of cubes as they pass 
through. Similarly the third wall of the Block will 
come into the plane, and looking at them similarly, as 
it were through an upright piece of glass, we write their 
names : 

Mars Mala Merces Mala Tyro Mala 

Ala Mala Cortis Mala Aer Mala 

Sector Mala Hama Mala Remus Mala 

Now, it is evident that these slabs stand at different 
times for different parts of the cubes. We can imagine 
them to stand for the Moena of each cube as it passes 
through. In that case, the first set of slabs, which we 
put up, represents the Moenas of the front wall of cubes ; 
the next set, the Moenas of the second wall. Thus, if 
all the three sets of slabs be together on the table, we 
have a representation of the sections of the cube. For 
some purposes it would be better to have four sets of 
slabs, the fourth set representing the Murex of the 
third wall ; for the three sets only show the front faces 
of the cubes, and therefore would not indicate anything 
about the back faces of the Block. For instance, if a 
line passed through the Block diagonally from the 
point Corvus (Gold) to the point Lama (Deep-blue), it 
would be represented on the slabs by a point at the 
bottom left-hand corner of the Gold slab, a second point 
at the same corner of the Light-buff slab, and a third 
at the same corner of the Deep-blue slab. Thus, we 
should have the points mapped at which the line entered 
the fronts of the walls of cubes, but not the point in 
Lama at which it would leave the Block. 

Let the Diagrams i, 2, 3 (Fig. 11), be the three sets 
of slabs. To see the diagrams properly, the reader must 
set the top of the page on the table, and look along the 
page from the bottom of it. The line in question, which 



144 A New Ei'a of Thought. 

runs from the bottom left-hand near corner to the top 
right-hand far corner of the Block will be represented in 
the three sets of slabs by the points A, B, C. To com- 
plete the diagram of its course, we need a fourth set of 
slabs for the Murex of the third wall ; the same object 
might be attained, if we had another Block of 27 cubes 
behind the first Block and represented its front or 
Moenas by a set of slabs. For the point, at which the 
line leaves the first Block is identical with that at which 
it enters the second Block. 

If we suppose a sheet of glass to be the plane-world, 
the Diagrams i, 2, 3 (Fig. 11), may be drawn more 
naturally to us as Diagrams a, /3, 7 (Fig. 12). Here a 
represents the Moenas of the first wall, jS those of the 
second, 7 those of the third. But to get the plane- 
being's view we must look over the edge of the glass 
down the Z axis. 

Set 2 of slabs represent the Moenas of Wall 2. These 
Moenas are in contact with the Murex of Wall i. Thus 
Set 2 will show where the line issues from Wall i as 
well as where it enters Wall 2. 

The plane-being, therefore, could get an idea of the 
Block of cubes by learning these slabs. He ought not 
to call the Gold slab Urna Mala, but Urna Moena, and 
so on, because all that he learns are Moenas, merely the 
thin faces of the cubes. By introducing the course of 
time, he can represent the Block more nearly. For, if 
he supposes it to be passing an inch a minute,- he may 
give the name Urna Mala to the Gold slab enduring for 
a minute. 

But, when he has learnt the slabs in this position and 
sequence, he has only a very partial view of the Block. 
Let the Block turn round the Z axis, as Model i turns 
round the Brown line. A different set of cubes comes 
into his plane, and now they come in on the Alvus 



Ar 



■>X 



Z 



0) 











6 











->x 



z 



f2J 



F,^ M. 



A 











B 











F.^IS 



"c 



i 



(^) 



>x 



,£ 



G (a) F /-/s; fy; 



\To facet- I44- 



Sfcfcr TTcnum Urna 



(I) 



M#rc« Mot* TibtctK 



O -X 



(z) 



Tyro 01i« Vlttil 



^3) 



F.g. 13 



-Z 



Urn-a 



-z 



-z 



MoUs &iltvs 



Oltrun Bi(l«ns 



5c(na 



Co77I« 



(I) 



(2) 



Vnlis. 



(3) 



[To face f. 145. 



Three-Space by Names, and in a Plane. 145 

faces. (AlvLis is here used to denote the left-hand faces 
of the cubes, and is not supposed to be Vermihon ; it is 
simply the thinnest slice on the left hand of the cube 
and of the same colour as the cube.) To represent this, 
the plane-being should employ a fresh set of slabs, for 
there is nothing common to the Moena and Alvus faces 
except an edge. But, since each cube is of the same 
colour throughout, the same slab may be used for its 
different faces. Thus the Alvus of Urna Mala can be 
represented by a Gold slab. Only it must never be 
forgotten that it is meant to be a new slab, and is not 
identical with the same slab used for Moena. 

Now, when the Block of cubes has turned round the 
Brown line into the plane, it is clear that they will be 
on the side of the Z axis opposite to that on which 
were the Moena slabs. The line, which ran Y, now runs 
—X. Thus the slabs will occupy the second quadrant 
marked by the axes, as shown in the diagram (Fig. 13). 
Each of these slabs we will name Alvus. In this view, 
as before, the book is supposed to be tilted up towards 
the reader, so that the Z axis runs from O to his eye. 
Then, if the Block be passed at right angles through the 
plane, there will come into view the two sets of slabs 
represented in the Diagrams (Fig. 13). In copying this 
arrangement with the slabs, the cardboard on which 
they are arranged must slant upwards to the eye, i.e., 
OZ must run up to the eye, and the sides of the slabs 
seen are in Diagram 2 (Fig. 13), the upper edges of 
Tibicen, Mora, Merces ; in Diagram 3, the upper edges 
of Vestis, Oliva, Tyro. 

There is another view of the Block possible to a plane- 
being. If the Block be turned round the X axis, the 
lower face comes into the vertical plane. This corre- 
sponds to turning Model i round the Orange line. On 
referring to the diagram (Fig. I4), we now see that the 

L 



146 A New Era of Thought. 

name of the faces of the cubes coming into the plane is 
Syce. Here the plane-being looks from the extremity 
of the Z axis and the squares, which he sees run from 
him in the — Z direction. (As this turn of the Block 
brings its Syce into the vertical plane so that it ex- 
tends three inches below the base line of its Moena, it 
is evident that the turn is only possible if the Moena be 
originally at a height of at least three inches above the 
plane-being's earth line in the vertical plane.) Next, if 
the Block be passed through the plane, the sections 
shown in the Diagrams 2 and 3 (Fig. 14) are brought 
into view. 

Thus, there are three distinct ways of regarding the 
cubic Block, each of them equally primary ; and if the 
plane-being is to have a correct idea of the Block, he 
must be equally familiar with each view. By means of 
the slabs each aspect can be represented ; but we must 
remember in each of the three cases, that the slabs 
represent different parts of the cube. 

When we look at the cube Pallor Mala in space, we 
see that it touches six other cubes by its six faces. But 
the plane-being could only arrive at this fact by com- 
paring different views. Taking the three Moena sec- 
tions of the Block, he can see that Pallor Mala Moena 
touches Plebs Moena, Mora Moena, Uncus Moena, and 
Tergum Moena by lines. And it takes the place of 
Bidens Moena, and is itself displaced by Cortis Moena 
as the Block passes through the plane. Next, this 
same Pallor Mala can appear to him as an Alvus. In 
this case, it touches Plebs Alvus, Mora Alvus, Bidens 
Alvus, and Cortis Alvus by lines, takes the place of 
Uncus Alvus, and is itself displaced by Tergum Alvus 
as the Block moves. Similarly he can observe the 
relations, if the Syce of the Block be in his plane. 

Hence, this unknown body Pallor Mala appears to 



Urna 



Ma1<« 
Moms 



BiJeu 
Mocnt 






Fi«f5 



■'5- 



-X 



Mol«« 
Mo«M 



BtdeiU 

Mociu 



TllcMK 



Utnt 

Mo/ti4 



%'& 



z 

[To /ace ^. 147. 



Three-Space by Names, and in a Plane. 147 

him now as one plane-figure now as another, and comes 
before him in different connections. Pallor Mala is that 
which satisfies all these relations. Each of them he can 
in turn present to sense ; but the total conception of 
Pallor Mala itself can only, if at all, grow up in his mind. 
The way for him to form this mental conception, is to 
go through all the practical possibilities which Pallor 
Mala would afford him by its various movements and 
turns. In our world these various relations are found 
by the most simple observations ; but a plane-being 
could only acquire them by considerable labour. And 
if he determined to obtain a knowledge of the physical 
existence of a higher world than his own, he must pass 
through such discipline. 



We will see what change could be introduced into the 
shapes he builds by the movements, which he does not 
know in his world. Let us build up this shape with the 
cubes of the Blocji : Urna Mala, Moles Mala, Bidens 
Mala, Tibicen Mala. To the plane-being this shape 
would be the slabs, Urna Moena, Moles Moena, Bidens 
Moena, Tibicen Moena (Fig. 15). Now let the Block 
be turned round the Z axis, so that it goes past the 
position, in which the Alvus sides enter the vertical 
plane. Let it move until, passing through the plane, 
the same Moena sides come in again. The mass of the 
Block will now have cut through the plane and be on the 
near side of it towards us ; but the Moena faces only will 
be on the plane-being's side of it. The diagram (Fig. 16) 
shows what he will see, and it will seem to him similar 
to the first shape (Fig. 15) in every respect except 
its disposition with regard to the Z axis. It lies in the 
direction —X, opposite to that of the first figure. How- 
ever much he turn these two figures about in the plane. 



148 A New Era of Thought. 

he cannot make one occupy the place of the other, part 
for part. Hence it appears that, if we turn the plane- 
being's figure about a line, it undergoes an operation 
which is to him quite mysterious. He cannot by any 
turn in his plane produce the change in the figure pro- 
duced by us. A little observation will show that a 
plane-being can only turn round a point. Turning 
round a line is a process unknown to him. Therefore 
one of the elements in a plane-being's knowledge of a 
space higher than his own, will be the conception of a 
kind of turning which will change his solid bodies into 
their own images. 



CHAPTER VI. 

THE MEANS BY WHICH A PLANE-BEING WOULD 
ACQUIRE A CONCEPTION OF OUR FIGURES. 

Take the Block of twenty-seven Mala cubes, and build 
up the following shape (Fig. i8) : — 

Urna Mala, Moles Mala, Plebs Mala, Pallor Mala:, 
Mora Mala. 

If this shape, passed through the vertical plane, the 
plane-being would perceive : — 

(i) The squares Urna Moena and Moles Moena. 

(2) The three squares Plebs Moena, Pallor Moena, 
Mora Moena, 

and then the whole figure would have passed through 
his plane. 

If the whole Block were turned round the Z axis till 
the Alvus sides entered, and the figure built up as it 
would be in that disposition of the cubes, the plane-being 
would perceive during its passage through the plane : — 

(i) Urna Alvus ; 

(2) Moles Alvus, Plebs Alvus, Pallor Alvus, Mora 
Alvus, which would all enter on the left side of the Z 
axis. 

Again, if the Block were turned round the X axis, the 
Syce side would enter, and the cubes appear in the 
following order :— 

(i) Urna Syce, Moles Syce, Plebs Syce ; 

(2) Pallor Syce ; 

(3) Mora Syce. 



15° 



A New Era of Thoiight. 



A comparison of these three sets of appearances would 
give the plane-being a full account of the figure. It is 
that which can produce these various appearances. 

Let us now suppose a glass plate placed in front of 
the Block in its first position. On this plate let the axes 
X and Z be drawn. They divide the surface into four 
parts, to which we give the following names (Fig. 17) : — 

ZX = that quarter defined by the positive Z and posi- 
tive X axis. 

ZX = that quarter defined by the positive Z and 
negative X axis (which is called " Z negative X "). 

ZX = that quarter defined by the negative Z and 
negative X axis. 

ZX = that quarter defined by the negative Z and 
positive X axis. 

The Block appears in these different quarters or quad- 
rants, as it is turned round the different axes. In Z X 
the Moenas appear, in Z X the Alvus faces, in Z X the 
In each quadrant are drawn nine squares, to 
the faces of the cubes when they enter. For 
instance, in Z X we have the Moenas of : — 

Z 

Comes Tibicen Vestis 
Ostrum Bidens Scena 
Urna Moles Saltus 
X 



Syces, 
receive 



And in Z X we have the Alvus of : 



Mars 


Spicula 


Ala 


Uncus- 


Sector 


Frenum 



-X- 



Comes 

Ostrum 

Urna 



And in the Z X we have the Syces of : — 



Urna 

Frenum 

Sector 



Moles 
Plebs 
Hama 



Saltus 
Sypho 
Remus 



-X 



Al 



vus 



Moend 



-X 



Rgy 



Jyces 



-Z 



[ To face ^. r'so. 



fl/ 



r~L 



-X 



(1) 






%'8 f^) 



f 



2 2 



-X 



r/) 



r^) 



r.^. 19 



(3) 



\Tiifocep. 151, 



Plane-Being' s Conception of our Figures. 151 

Now, if the shape taken at the beginning of this chapter 
be looked at through the glass, and the distance of the 
second and third walls of the shape behind the glass 
be considered of no account — that is, if they be treated 
as close up to the glass — we get a plane outline, which 
occupies the. squares Urna Moena, Moles Moena, Bidens 
Moena, Tibicen Moena. This outline is called a pro- 
jection of the figure. To see it like a plane-being, we 
should have to look down on it along the Z axis. 

It is obvious that one projection does not give the 
shape. For instance, the square Bidens Moena might 
be filled by either Pallof or Cortis. All that a square in 
the room of Bidens Moena denotes, is that there is a 
cube somewhere in the Y, or unknown, direction from 
Bidens Moena. This view, just taken, we should call 
the front view in our space ; we are then looking at it 
along the negative Y axis. 

When the same shape is turned round on the Z axis, 
so as to be projected on the Z X quadrant, we have the 
squares — Urna Alvus, Frenum Alvus, Uncus Alvus, 
Spicula Alvus. When it is turned round the X axis, 
and projected on Z X, we have the squares, Urna Syce, 
Moles Syce, Plebs Syce, and no more. This is what is 
ordinarily called the ground plan ; but we have set it in a 
different position from that in which it is usually drawn. 

Now, the best method for a plane-being of familiar- 
izing himself with shapes in our space, would be to 
practise the realization of them from their different pro- 
jections in his plane. Thus, given the three projections 
just mentioned, he should be able to construct the figure 
from which they are derived. The projections (Fig. 19) 
are drawn below the perspective pictures of the shape 
(Fig. 18). From the front, or Moena view, he would 
conclude that the shape was Urna Mala, Moles Mala, 
Bidens Mala, Tibicen Mala ; or instead of these, or also 



152 A New Era of Thought. 

in addition to them, any of the cubes running in the Y 
direction from the plane. That is, from the Moena pro- 
jection he might infer the presence of all the following 
cubes (the word Mala is omitted for brevity) : Urna, 
Frenum, Sector, Moles, Plebs, Hama, Bidens, Pallor, 
Cortis, Tibicen, Mora, Merces. 

Next, the Alvus view or projection might be given by 
the cubes (the word Mala being again omitted) : Urna, 
Moles, Saltus, Frenum, Plebs, Sypho, Uncus, Pallor, 
Tergum, Spicula, Mora, Oliva Lastly, looking at the 
ground plan or Syce view, he would infer the possible 
presence of Urna, Ostrum, Comes, Moles, Bidens, 
Tibicen, Plebs, Pallor, Mora. 

Now, the shape in higher space, which is usually there, 
is that which is common to all these three appearances. 
It can be determined, therefore, by rejecting those cubes 
which are not present in all three lists of cubes possible 
from the projections. And by this process the plane- 
being could arrive at the enumeration of the cubes 
which belong to the shape of which he had the pro- 
jections. After a time, when he had experience of the 
cubes (which, though invisible to him as wholes, he 
could see part by part in turn entering his space), the 
projections would have more meaning to him, and he 
might comprehend the shape they expressed fragmen- 
tarily in his space. To practise the realization from 
projections, we should proceed in this way. First, we 
should think of the possibilities involved in the Moena 
view, and build them up in cubes before us. Secondly, 
we should build up the cubes possible from the Alvus 
view. Again, taking the shape at the beginning of the 
chapter, we should find that the shape of the Alvus 
possibilities intersected that of the Moena possibilities in 
Urna, Moles, Frenum, Plebs, Pallor, Mora ; or, in other 
words, these cubes are common to both. Thirdly, we 



Plane- Beings Conception of our Figures. 153 

should build up the Syce possibilities, and, comparing 
their shape with those of the Moena and Alvus pro- 
jections, we should find Urna, Moles, Plebs, Pallor, Mora, 
of the Syce view coinciding with the same cubes of the 
other views, the only cube present in the intersection of 
the Moena and Alvus possibilities, and not present in 
the Syce view, being Frenum. 

The determination of the figure denoted by the three 
projections, may be more easily effected by treating each 
projection as an indication of what cubes are to be cut 
away. Taking the same shape as before, we have in the 
Moena projection Urna, Moles, Bidens, Tibicen ; and 
the possibilities from them are Urna, Frenum, Sector, 
Moles, Plebs, Hama, Bidens, Pallor, Cortis, Tibicen, 
Mora, Merces. This may aptly be called the Moena 
solution. Now, from the Syce projection, we learn at 
once that those cubes, which in it would produce Frenum, 
Sector, Hama, Remus, Sypho, Saltus, are not in the 
shape. This absence of Frenum and Sector in the Syce 
view proves that their presence in the Moena solution is 
superfluous. The absence of Hama removes the possi- 
bility of Hama, Cortis, Merces. The absence of Remus, 
Sypho, Saltus, makes no difference, as neither they nor 
any of their Syce possibilities are present in the Moena 
solution. Hence, the result of comparison of the Moena 
and Syce projections and possibilities is the shape : 
Urna, Moles, Plebs, Bidens, Pallor, Tibicen, Mora. This 
may be aptly called the Moena-Syce solution. Now, 
in the Alvus projection we see that Ostrum, Comes, 
Sector, Ala, and Mars are absent. The absence of 
Sector, Ala, and Mars has no effect on our Moena-Syce 
solution ; as it does not contain any of their Alvus possi- 
bilities. But the absence of Ostrum and Comes proves 
that in the Moena-Syce solution Bidens and Tibicen are 
superfluous, since their presence in the original shape 



154 A New Era of Thought. 

would give Ostrum and Comes in the Alvus projection. 
Thus we arrive at the Moena-Alvus-Syce solution, 
which gives us the shape : Urna, Moles, Plebs, Pallor, 
Mora. 

It will be obvious on trial that a shape can be instantly 
recognised from its three projections, if the Block be 
thoroughly well known in all three positions. Any 
difficulty in the realization of the shapes comes from the 
arbitrary habit of associating the cubes with some one 
direction in which they happen to go with regard to us. 
If we remember Ostrum as above Urna, we are not 
remembering the Block, but only one particular relation 
of the Block to us. That position of Ostrum is a fact 
as much related to ourselves as to the Block. There is, 
of course, some information about the Block implied in 
that position ; but it is so mixed with information about 
ourselves as to be ineffectual knowledge of the Block. 
It is of the highest importance to enter minutely into 
all the details of solution written above. For, corre- 
sponding to every operation necessary to a plane-being 
for the comprehension of our world, there is an opera- 
tion, with which we have to become familiar, if in our 
turn we would enter into some comprehension of a 
world higher than our own. Every cube of the Block 
ought to be thoroughly known in all its relations. And 
the Block must be regarded, not as a formless mass out 
of which shapes can be made, but as the sum of all 
possible shapes, from which any one we may choose is a 
selection. In fact, to be familiar with the Block, we 
ought to know every shape that could be made by any 
selection of its cubes ; or, in other words, we ought to 
make an exhaustive study of it. In the Appendix is 
given a set of exercises in the use of these names (which 
form a language of shape), and in various kinds of pro- 
jections. The projections studied in this chapter are 



P lane- Being s Conception of our Figures. 155 

not the only, nor the most natural, projections by which 
a plane-being would study higher space. But they 
suffice as an illustration of our present purpose. If the 
reader will go through the exercises in the Appendix, 
and form others for himself, he will find every bit of 
manipulation done will be of service to him in the com- 
prehension of higher space. 

There is one point of view in the study of the Block, 
by means of slabs, which is of some interest. The cubes 
of the Block, and therefore also the representative slabs 
of their faces, can be regarded as forming rows and 
columns. There are three sets o? them. If we take 
the Moena view, they represent the views of the three 
walls of the Block, as they pass through the plane. To 
form the Alvus view, we only have to rearrange the 
slabs, and form new sets. The first new set is formed 
by taking the first, or left-hand, column of each of the 
Moena sets. The second Alvus set is formed by taking 
the second or middle cojumns of the three Moena sets. 
The third will consist of the remaining or right-hand 
columns of the Moenas. 

Similarly, the three Syce sets may be formed from 
the three horizontal rows or floors of the Moena sets. 

Hence, it appears that the plane-being would study 
our space by taking all the possible combinations of the 
corresponding rows and columns. He would break up 
the first three sets- into other sets, and the study of the 
Block would practically become to him the study of 
these various arrangements. 



CHAPTER VII. 

FOUR-SPACE : ITS REPRESENTATION IN THREE- 
SPACE. 

We now come to the essential difficulty of our task 
All that has gone before is preliminary. We have now 
to frame the method by which we shall introduce 
through our space-figures the figures of a higher space. 
When a plane-being studies our shapes of cubes, he has 
to use squares. He is limited at the outset. A cube 
appears to him as a square. On Model i we see the 
various squares as which the cube can appear to him. 
We suppose the plane-being to look from the extremity 
of the Z axis down a vertical plane. First, there is the 
Mbena square. Then there is the square given by a 
section parallel to Moena, which he recognises by the 
variation of the bounding lines as soon as the cube 
begins to pass through his plane. Then comes the 
Murex square. Next, if the cube be turned round the 
Z axis and passed through, he sees the Alvus and Proes 
squares and the intermediate section. So too with the 
Syce and Mel squares and the section between them. 

Now, dealing with figures in higher space, we are in 
an analogous position. We cannot grasp the element 
of which they are composed. We can conceive a cube ; 
but that which corresponds to a cube in higher space is 
beyond our grasp. But the plane-being was obliged to 
use two-dimensional figures, squares, in arriving at a 
notion of a three-dimensional figure ; so also must we 



Representation of Four- Space. 157 

use three-dimensional figures to arrive at the notion of 
a four-dimensional. Let us call the figure which corre- 
sponds to a square in a plane and a cube in our space, a 
tessaract. Model i is a cube. Let us assume a tessa- 
ract generated from it. Let us call the tessaract Urna. 
The generating cube may then be aptly called Urna Mala. 
We may use cubes to represent parts of four-space, but 
we must always remember that they are to us, in our 
study, only what squares are to a plane-being with re- 
spect to a cube. 

Let us again examine the mode in which a plane- 
being represents a Block of cubes with slabs. Take 
Block I of the 81 Set of cubes. The plane-being repre- 
sents this by nine slabs, which represent the Moena face 
of the block. Then, omitting the solidity of these first 
nine cubes, he takes another set of nine slabs to repre- 
sent the next wall of cubes. Lastly, he represents the 
third wall by a third set, omitting the solidity of both 
second and third walls. In this manner, he evidently 
represents the extension of the Block upwards and side- 
ways, in the Z and X directions ; but in the Y direction 
he is powerless, and is compelled to represent extension 
in that direction by setting the three sets of slabs 
alongside in his plane. The second and third sets de- 
note the height and breadth of the respective walls, but 
not their depth or thickness. Now, note that the Block 
extends three inches in each of the three directions. 
The plane-being can represent two of these dimensions 
on his plane ; but the unknown direction he has to 
represent by a repetition of his plane figures. The cube 
extends three inches in the Y direction. He has to use 
3 sets of slabs. 

The Block is built up arbitrarily in this manner ; 
Starting from Urna Mala and going up, we come to a 
Brown cube, and then to a Light-blue cube. Starting 



158 A New Era of Thought. 

from Urna Mala and going right, we come to an Orange 
and a Fawn cube. Starting from Urna Mala and going 
away from us, we come to a Blue and a Buff cube. 
Now, the plane-being represents the Brown and Orange 
cubes by squares lying next to the square which repre- 
sents Urna Mala. The Blue cube is as close as the 
Brown cube to Urna Mala, but he can find no place in 
the plane where he can place a Blue square so as to 
show this co-equal proximity of both cubes to the first. 
So he is forced to put a Blue square anywhere in his 
plane and say of it : " This Blue square represents what 
I should arrive at, if I started from Urna Mala and 
moved away, that is in the Y or unknown direction." 
Now, just as there are three cubes going up, so there 
are three going away. Hence, besides the Bl^e square 
placed anywhere on the plane, he must also place a Buff 
square beyond it, to show that the Block extends as far 
away as it does upwards and sideways. (Each cube 
being a different colour, there will be as many different 
colours of squares as of cubes.) It will easily be seen 
that not only the Gold square, but also the Orange and 
every other square in the first set of slabs must have two 
other squares set somewhere beyond it on the plane to 
represent the extension of the Block away, or in the 
unknown Y direction. 

Coming now to the representation of a four-dimen- 
sional block, we see that we can show only three dimen- 
sions by cubic blocks, and that the. fourth can only be 
represented by repetitions of such blocks. There must 
be a certain amount of arbitrary naming and colouring. 
The colours have been chosen as now stated. Take the 
first Block of the 81 Set. We are familiar vi^ith its 
colours, and they can be found at any time by reference 
to Model I. Now, suppose the Gold cube to represent 
what we can see in our space of a Gold tessaract ; the 



Representation of Four-Space. 159 

other cubes of Block i give the colours of the tessaracts 
which lie in the three directions X, Y, and Z from the 
Gold one. But what is the colour of the tessaract which 
lies next to the Gold in the unknown direction, W? 
Let us suppose it to be Stone in colour. Taking out 
Block 2 of the 81 Set and arranging it on the pattern of 
Model 9, we find in it a Stone cube. But, just as there are 
three tessaracts in the X, Y, and Z directions, as shown 
by the cubes in Block i, so also must there be three 
tessaracts in the unknown direction, W. Take Block 3 
of the 81 'Set. This Block can be arranged on the 
pattern of Model 2. In it there is a Silver cube where 
the Gold cube lies in Block i. Hence, we may say, the 
tessaract which comes next to the Stone one in the 
unknown direction from the Gold, is of a Silver colour. 
Now, a cube in all these cases represents a tessaract. 
Between the Gold and Stone cubes there is an inch in 
the unknown direction. The Gold tessaract is supposed 
to be Gold throughout in all four directions, and so also 
is the Stone. We may imagine it in this way. Sup- 
pose the set of three tessaracts, the Gold, the Stone, and 
the Silver to move through our space at the rate of an 
inch a minute. We should first ■ see the _ Gold cube 
which would last a minute, then the Stone cube for a 
minute, and lastly the Silver cube a minute. (This is 
precisely analogous to the appearance of passing cubes 
to the plane-being as successive squares lasting a 
minute.) After that, nothing would be visible. 

Now, just as we must suppose that there are three 
tessaracts proceeding from the Gold cube in the un- 
known direction, so there must be three tessaracts ex- 
tending in the unknown direction from every one of the 
27 cubes of the first Block. The Block of 27 cubes is 
not a Block of 27 tessaracts, but it represents as much 
of them as we can see at once in our space ; and they 



i6o A New Era of Thought. 

form the first portion or layer (like the first wall of 
cubes to the plane-being) of a set of eighty-one tessa- 
racts, extending to equal distances in all four directions. 
Thus, to represent the whole Block of tessaracts there 
are 8 1 cubes, or three Blocks of 27 each. 

Now, it is obvious that, just as a cube has various 
plane boundaries, so a tessaract has various cube bound- 
aries. The cubes of the tessaract, which we have been 
regarding, have been those containing the X, Y, and Z 
directions, just as the plane-being regarded the Moena 
faces containing the X and Z directions. And, as long 
as the tessaract is unchanged in its position with regard 
to our space, we can never see any portion of it which 
is in the unknown direction. Similarly, we saw that a 
plane-being could not see the parts of a cube which went 
in the third direction, until the cube was turned round 
one of its edges. In order to make it quite clear what 
parts of a cube came into the plane, we gave distinct 
names to them. Thus, the squares containing the Z and 
X directions were called Moena and Murex ; those con- 
taining the Z and Y, Alvus and Proes ; and those the 
X and Y, Syce and Mel. Now, similarly with our four 
axes, any three will determine a cube. Let the tessaract 
in its normal position have the cube Mala determined by 
the axes Z, X, Y. Let the cube Lar be that which is 
determined by X, Y, W, that is, the cube which, starting 
from the X Y plane, stretches one inch in the unknown 
or W direction. Let Vesper be the cube determined by 
Z, Y, W, and Pluvium by Z, X, W. And let these cubes, 
have opposite cubes of the following names : 

Mala has Margo 

Lar „ Velum 

Vesper „ Idus 

Pluvium „ Tela' 

Another way of looking at the matter is this. When 



Repyesenlation of Four- Space. i6i 

a cube is generated from a square, each of the lines 
bounding the square becomes a square, and the square 
itself becomes a cube, giving two squares in its initial 
and final positions. When a cube moves in the new 
and unknown direction, each of its planes traces a cube 
and it generates a tessaract, giving in its initial and 
final positions two cubes. Thus there are eight cubes 
bounding the tessaract, six of them from the six plane 
sides and two from the cube itself These latter two 
are Mala and Margo. The cubes from the six sides are : 
Lar from Syce, Velum from Mel, Vesper from Alvus, 
Idus from Proes, Pluvium from Moena, Tela from Murex. 
And just as a plane-being can only see the squares of a 
cube, so we can only see the cubes of a tessaract. It 
may be said that the cube can be pushed partly through 
the plane, so that the plane-being sees a section between 
Moena and Murex. Similarly, the tessaract can be 
pushed through our space so that we can see a section 
between Mala and Margo. 

There is a method of approaching the matter, which 
settles all difficulties, and provides us with a nomencla- 
ture for every part of the tessaract. We have seen how 
by writing down the names of the cubes of a block, and 
then supposing that their number increases, certain sets 
of the names come to denote points, lines, planes, and 
solid. Similarly, if we write down a set of names of 
tessaracts in a block, it will be found that, when their 
number is increased, certain sets of the names come to 
denote the various parts of a tessaract. 

For this purpose, let us take the 8i Set, and use the 
cubes to represent tessaracts. The whole of the 8i 
cubes make one single tessaractic set extending three 
inches in each of the four directions. The names must 
be remembered to denote tessaracts. Thus, Corvus is a 
tessaract which has the tessaracts Cuspis and Nugse to 

M 



i62 A New Era of Thoug^ht. 

the right, Arctos and Ilex above it, Dos and Cista away 
from it, and Ops and Spira in the fourth or unknown 
direction from it. It will be evident at once, that to 
write these names in any representative order we must 
adopt an arbitrary system. We require them running 
in four directions ; we have only two on paper. The X 
direction (from left to right) and the Y (from the bottom 
towards the top of the page) will be assumed to be truly 
represented. The Z direction will be symbolized by 
writing the names in floors, the upper floors always 
preceding the lower. Lastly, the fourth, or W, direction 
(which has to be symbolized in three-dimensional space 
by setting the solids in an arbitrary position) will be 
signified by writing the names in blocks, the name which 
stands in any one place in any block being next in the 
W direction to that which occupies the same position in 
the block before or after it. Thus, Ops is written in the 
same place in the Second Block, Spira in the Third 
Block, as Corvus occupies in the First Block. 

Since there are an equal number of tessaracts in each 
of the four directions, there will be three floors Z when 
there are three X arid Y. Also, there will be three 
Blocks W. If there be four tessaracts in each direction, 
there will be four floors Z, and four blocks W. Thus, 
when the number in each direction is enlarged, the 
number of blocks W is equal to the number of tessaracts 
in each known direction. 

On pp. 136, 137 were given the names as used for a 
cubic block of 27 or 64. Using the same and more 
names for a tessaractic Set, and remembering that each 
name now represents, not a cube, but a tessaract, we 
obtain the following nomenclature, the order in which 
the names are written being that stated above : 



Representation of Four-Space. 



163 



. r Solia 

PP f I Lensa 
^'°°''- Uelis 



Third Block. 

Livor' 
Lares 
Tholus 



Middle 
Floor. 



Lower 
Floor. 



Lixa 
Crux 

Pagus 



( Panax 
\ Opex 
(. Spira 



Portica 
Margo 
Silex 



Mensura 

Lappa 

Luca 



Talus 
Calor 
Passer 

Vena 

Sal 

Onager 

Mugil 

Mappa 

Ancilla 



Upper 
Floor. 



Orsa 
Creta 
Lucta 



Second Block. 

Mango 
Velum 
Limbus 



Middle 
Floor, 



:1 



Camoena 

Vesper 

Pagina 



Lower J ^ 
^'""""•lops 



Tela 

Tessaract 

Pluvium 



Lorica 

Lar 

Lotus 



Libera 

Meatus 

Pator 

Orca 

Idus 
Pactum 

Offex 

011a 

Limus 



UPP^"- Via 
Fl""--- (.Ilex 



First Block. 

Semita 

Mel 

Callis 



Bucina 
, AIvus 
■ ( Arctos 



Middle ( 



f Cista 
Lower U„^ 

^^°°''- (Corvus 



Murex 

Mala 

Moena 



Cadus 

Syce 

CuSpis 



Lama 

Iter 

Sors 

Daps 
Proes 
Far 

Crus 
Bolus 

Nuga? 



164 A New Era of Thought. 

It is evident that this set of tessaracts could be 
increased to the number of four in each direction, 
the names being used as before for the cubic blocks 
on pp. 136, 137, and in that case the Second Block 
would be duplicated to make the four blocks required 
in the unknown direction. Comparing such an 81 Set 
and 256 Set, we should find that Cuspis, which was 
a single tessaract in the 81 Set became two tessaracts 
in the 256 Set. And, if we introduced a larger number, 
it would simply become longer, and not increase in 
any other dimension. Thus, Cuspis would become the 
name of an edge. Similarly, Dos would become the 
name of an edge, and also Arctos. Ops, which is found 
in the Middle Block of the 81 Set, occurs both in the 
Second and Third Blocks of the 256 Set ; that is, it also 
tends to elongate and not extend in any other direction, 
and may therefore be used as the name of an edge of 
a tessaract. 

Looking at the cubes which represent the Syce tessar- 
acts, we find that, though they increase in number, they 
increase only in two directions ; therefore, Syce may be 
taken to signify a square. But, looking at what comes 
from Syce in the W direction, we find in the Middle 
Block of the 81 Set one Lar, and in the Second and 
Third Blocks of the 256 Set four Lars each. Hence, Lar 
extends in three directions, X, Y, W, and becomes a cube. 
Similarly, Moena is a plane; but Pluvium, which proceeds 
from it, extends not only sideways and upwards like 
Moena, but in the unknown direction also. It occurs 
in both Middle Blocks of the 256 Set. Hence, it also 
is a cube. We have now considered such parts of the 
Sets as contain one, two, and three dimensions. But 
there is one part which contains four. It is that named 
Tessaract. In the 256 Set there are eight such cubes in 
the Second, and eight in the Third Block ; that is, they 



Representation of Four-Space. 165 

extend Z, X, Y, and also W. They may, therefore, be 
considered to represent that part of a tessaract or 
tessaractic Set, which is analogous to the interior of a 
cube. 

The arrangement of colours corresponding to these 
names is seen on Model i corresponding to Mala, Model 
2 to Margo, and Model 9 to the intermediate block. 

When we take the view of the tessaract with which 
we commenced, and in which Arctos goes Z, Cuspis X, 
Dos Y, and Ops W, we see Mala in our space. But 
when the tessaract is turned so that the Ops line goes 
- X, while Cuspis is turned W, the other two remaining 
as they were, then we do not see Mala, but that cube 
which, in the original position of the tessaract, contains 
the Z, Y, W, directions, that is, the Vesper cube. 

A plane-being may begin to study a block of cubes 
by their Syce squares ; but if the block be turned round 
Dos, he will have Alvus squares in his space, and he 
must then use them to represent the cubic Block. So, 
when the tessaractic Set is turned round. Mala cubes 
leave our space, and Vespers enter. 

There are two ways which can be followed in studying 
the Set of tessaracts. 

I. Each tessaract of one inch every way can be 
supposed to be of the same colour throughout, so that, 
whichever way it be turned, whichever of its edges 
coincide with our known axes, it appears to us as a cube 
of one uniform colour. Thus, if Urna be the tessaract, 
Urna Mala would be a Gold cube, Urna Vesper a Gold 
cube, and so on. This method is, for the most part, 
adopted in the following pages. In this case, a whole 
Set of 4x4x4x4 tessaracts would in colours resemble 
a set composed of four cubes like Models i, 9, 9, and 2. 
But, when any question about a particular tessaract has 
to be settled, it is advantageous, for the sake of distinct- 



1 66 A New Era of Thought. 

ness, to suppose it coloured in its diiferent regions as 
the whole set is coloured. 

II. The other plan is, to start with the cubic sides 
of the inch tessaract, each coloured according to the 
scheme of the Models i to 8. In this case, the lines, if 
shown at all, should be very thin. For, in fact, only 
the faces would be seen, as the width of the lines would 
only be equal to the thickness of our matter in the 
fourth dimension, which is indistinguishable to the 
senses. If such completely coloured cubes be used, less 
error is likely to creep in ; but it is a disadvantage that 
each cube in the several blocks is exactly like the others 
in that block. If the reader make such a set to work 
with for a time, he will gain greatly, for the real way of 
acquiring a sense of higher space is to obtain those 
experiences of the senses exactly, which the observation 
of a four-dimensional body would give. These Models 
1-8 are called sides of the tessaract. 

To make the matter perfectly clear, it is best to sup- 
pose that any tessaract or set of tessaracts which we 
examine, has a duplicate exactly similar in shape and 
arrangement of parts, but different in their colouring. 
In the first, let each tessaract have one colour through- 
out, so that all its cubes, apprehended in turn in our 
space, will be of one and the same colour. In the 
duplicate, let each tessaract be so coloured as to show 
its different cubic sides by their different colours. 
Then, when we have it turned to us in different aspects, 
we shall see different cubes, and when we try to trace 
the contacts of the tessaracts with each other, we shall 
be helped by realizing each part of every tessaract in 
its own colour. 



CHAPTER VIII. 

REPRESENTATION OF FOUR-SPACE BY NAME. 
STUDY OF TESSARACTS. 

We have now surveyed all the preliminary ground, and 
can study the masses of tessaracts without obscurity. 

We require a scaffold or framework for this purpose, 
which in three dimensions will consist of eight cubic 
spaces or octants assembled round one point, as in two 
dimensions it consisted of four squares or quadrants 
round a point. 

These eight octants lie between the three axes Z, X, 
Y, which intersect at the given point, and can be named 
according to their positions between the positive and 
negative directions of those axes. Thus the octant 
Z, X, Y, is that which is contained by the positive por- 
tions of all three axes ; the octant Z, X, Y, that which 
is to the left of Z, X, Y, and between the positive parts 
of Z and Y and the negative of X. To illustrate this 
quite clearly, let us take the eight cubes — Urna, Moles, 
Plebs, Frenum, Uncus, Pallor, Bidens, Ostrum — and 
place them in the eight octants. Let them be placed 
round the point of intersection of the axes ; Pallor 
Corvus, Plebs Ilex, etc., will be at that point. Their 
positions will then be : — 

Urna in the Octant ZXY 



Moles „ 


ZXY 


Plebs „ 


, ZXY 


Frenum „ 


ZXY 


Uncus „ 


, ZXY 


Pallor „ 


, ZXY 


Bidens ,, 


, ZXY 


Ostrum „ 


, ZXY 



167 



i68 



A New Era of Thought. 



The names used for the cubes, as they are before us, 
are as follows : — 



Arcus Mala 
_, , Laurus Mala 
^^°°''- Uxis Mala 



Third ' 



„ , i Postis Mala 

_. < Orcus Mala 
°°^' \ Verbum Mala 

_. f Telutn Mala 

Floor \ ^°'"^ "^^^^ 
■ \ Cervix Mala 



Third Block. 

Ovis Mala 
Tigris Mala 
Troja Mala 

Clipeus Mala 
Lacerta Mala 
Luctus Mala 

Nepos Mala 
Penates Mala 
Securis Mala 



Portio Mala 
Segmen Mala 
Aries Mala 

Tabula Mala 
Testudo Mala 
Anguis Mala 

Angusta Mala 
Vulcan Mala 
Vinculum Mala 



r Ara Mala 
^, \ Praeda Mala 
^^°°''- I Cortex Mala 



Third 



Second (P""™ Mala 

Floor. ) °"^^ M^l^ 
<. Cardo Mala 



Second Block. 

Vomer Mala 
Sacerdos Mala 
Mica Mala 

Glans Mala 
Tessera Mala 
Cudo Mala 



Pluma Mala 
Hydra Mala 
Flagellum Mala 

Colus Mala 
Domitor Mala 
Malleus Mala 



First f Agnien Mala 

Floor. \ C^^'e^ M''^^^ 
(.Thyrsus Mala 



Lacus Mala 
Cura Mala 
Vitta Mala 



Arvus Mala 
Limen Mala 
Sceptrum Mala 



Third S ^'"■' ^'""'^ 
Floor. 1 Spicula Mala 
V- Comes Mala 

Second (Ala Mala 
Floor. ) Uncus Mala 
I Ostrum Mala 

First f ■S^'^'o"' Mala 
Floor. ) Frenum Mala 
I Urna Mala 



First Block. 

Merces Mala 
Mora Mala 
Tibicen Mala 

Cortis Mala 
Pallor Mala 
Bidens Mala 

Hama Mala 
Plebs Mala 
Moles Mala 



Tyro Mala 
Oliva Mala 
Vestis Mala 

Aer Mala 
Tergum Mala 
Scena Mala 

Remus Mala 
Sypho Mala 
Saltus Mala 



Representation of Four-Space by Name. 169 

Their colours can be found by reference to the 
Models I, 9, 2, which correspond respectively to the 
First, Second, and Third Blocks. Thus, Urna Mala is 
Gold ; Moles, Orange ; Saltus, Fawn ; Thyrsus, Stone ; 
Cervix, Silver. The cubes whose colours are not shown 
in the Models, are Pallor Mala, Tessera Mala, and 
Lacerta Mala, which are equivalent to the interiors 
of the Model cubes, and are respectively Light-buff, 
Wooden, and Sage-green. These 81 cubes are the cubic 
sides and sections of the tessaracts of an 8 1 tessaractic 
Set, which measures three inches in every direction. 
We suppose it to pass through our space. Let us call 
the positive unknown direction Ana {i.e., -t-W) and the 
negative unknown direction Kata (— W). Then, as the 
whole tessaract moves Kata at the rate of an inch a 
minute, we see first the First Block of 27 cubes for one 
minute, then the Second, and lastly the Third, each 
lasting one minute. 

Now, when the First Block stands in the normal 
position, the edges of the tessaract that run from the 
Corvus corner of Urna Mala, are : Arctos in Z, Cuspis 
in X, Dos in Y, Ops in W. Hence, we denote this 
position by the following symbol : — 

Z X Y W 
a c d 

where a stands for Arctos, c for Cuspis, d for Dos, 
and for Ops, and the other letters for the four axes in 
space, a, c, d, are the axes of the tessaract, and can 
take up different directions in space with regard to us. 



Let us now take a smaller four-dimensional set. Of 
the 81 Set let us take the following : — 

Z X Y W 

a c d 



170 A New Era of Thought. 

Second Block. 

Co-„ J Tri« f Ocrea Mala Tessera Mala 

Second Floor.j^^^^^ ^^^^ ^^^^ ^^^^ 

First Floor, j^f " ^.^^ " ^T'^^f^ 

\Thyrsus Mala Vitta Mala 



Second Floor. -^ 



First Block. 

Uncus Mala Pallor Mala 

strum Mala Bidens Mala 



■c- u ■c•^ rFrenum Mala Plebs Mala 

First Floor, i ., ,, , ,, , i,t 1 

LUrna Mala Moles Mala 

Let the First Block be put up before us in Z X Y, 
(Urna Corvus is at the junction of our axes Z X Y). 
The Second Block is now one inch distant in the un- 
known direction ; and, if we suppose the tessaractic 
Set to move through our space at the rate of one 
inch a minute, the Second will enter in one minute, and 
replace the first. But, instead of this, let us suppose 
the tessaracts to turn so that Ops, which now goes W. 
shall go —X. Then we can see in our space that cubic 
side of each tessaract which is contained by the lines 
Arctos, Dos, and Ops, the cube Vesper ; and we shall 
no longer have the Mala sides but the Vesper sides of 
the tessaractic Set in our space. We will now build 
it up in its Vesper view (as we built up the cubic Block 
in its Alvus view). Take the Gold cube, which now 
means Urna Vesper, and place it on the left hand of its 
former position as Urna Mala, that is, in the octant 
Z X Y. Thyrsus Vesper, which previously lay just 
beyond Urna Vesper in the unknown direction, will 
now lie just beyond it in the —X direction, that is, 
to the left of it. The tessaractic Set is now in the 

.,. ZXYW,,, . . ^, 

position - , (the minus sign over the meaning 



'Ocrra /Uncw 



VesbCTS 



C^rio 



TTiyTS 



Oshi 



Urnj 



Unaiil Pillor' 



Ofitivm 



Urni 



Bidina 



MoUi 



M&la« 



%-20. 



[To/ace p. lyi' 



Representation of Four-Space by Name. 171 

that Ops runs in the negative direction), and its Vespers 
lie in the following order : — 

Second Block. 

fTessara Pallor 






Second Floor.-. „ , „. , 

ICudo Bidens 

T,. ^ T-i rCura Plebs 

First Floor. \^^. ,, , 

IVitta Moles 



First Block. 

)crea Uncus 



. roc 

■\Ca 



Second Floor.-, „ , ^ ^ 

(.Cardo Ostrum 

_. _, (■ Crates Frenum 

First Floor. -^ „, , . 

LThyrsus Urna 

The name Vesper is left out in the above list for the 
sake of brevity, but should be used in studying the 
positions. 

On comparing the two lists of the Mala view and 
Vesper view, it will be seen that the cubes presented in 
the Vesper view are new sides of the tessaract, and that 
the arrangement of them is different from that in the 
Mala view. (This is analogous to the changes in the 
•slabs from the Moena to Alvus view of the cubic Block.) 
Of course, the Vespers of all these tessaracts are not 
visible at once in our space, any more than are the 
Moenas of all three walls of a cubic Block to a plane- 
being. But if the tessaractic Set be supposed to move 
through space in the unknown direction at the rate 
of an inch a minute, the Second Block will present 
its Vespers after the First Block has lasted a minute. 
The relative position of the Mala Block and the Vesper 
Block may be represented in our space as in the dia- 
gram, Fig. 20. But it must be distinctly remembered 
that this arrangement is quite conventional, no more 
real than a plane-being's symbolization of the Moena 



172 A New Era of Thought. 

Wall and the Alvus Wall of the cubic Block by the 
arrangement of their Moena and Alvus faces, with the 
solidity omitted, along one of his known directions. 

The Vespers of the First and Second Blocks cannot 
be in our space simultaneously, any more than the 
Moenas of all three walls in plane space. To render 
their simultaneous presence possible, the cubic or 
tessaractic Block or Set must be broken up, and its 
parts no longer retain their relations. This fact is of 
supreme importance in considering higher space. End- 
less fallacies creep in as soon as it is forgotten that the 
cubes are merely representative as the slabs were, and 
the positions in our space merely conventional and 
symbolical, like those of the slabs along the plane. 
And these fallacies are so much fostered by again sym- 
bolizing the cubic symbols and their symbolical positions 
in perspective drawings or diagrams, that the reader 
should surrender all hope of learning space from this 
book or the drawings alone, and work every thought 
out with the cubes themselves. 

If we want to see what each individual cube of the 
tessaractic faces presented to us in the last example is 
like, we have only to consider each of the Malas simi-' 
lar in its parts to Model i, and each of the Vespers to 
Model 5. And it must always be remembered that the 
cubes, though used to represent both Mala and Vesper 
faces of the tessaract, mean as great a difference as the 
slabs used for the Moena and Alvus faces of the cube. 

If the tessaractic Set move Kata through our space, 
when the Vesper faces are presented to us, we see the 
following parts of the tessaract Urna (and, therefore, 
also the same parts of the other tessaracts) : 

(i) Urna Vesper, which is Model 5. 

(2) A parallel section between Urna Vesper and Urna 
Idus, which is Model 11. 



Representation of Four-Space by Name. 173 

(3) Urna Idus, which is Model 6. 

When Urna Idus has passed Kata our space, Moles 
Vesper enters it ; then a section between Moles Vesper 
and Moles Idus, and then Moles Idus. Here we have 
evidently observed the tessaract more minutely ; as it 
passes Kata through our space, starting on its Vesper 
side, we have seen the parts which would be generated 
by Vesper moving along Cuspis — that is Ana. 

Two other arrangements of the tessaracts have to be 
learnt besides those from the Mala and Vesper aspect. 
One of them is the Pluvium aspect. Build up the Set 
in Z X Y, letting Arctos run Z, Cuspis X, and Ops Y. 
In the common plane Moena, Urna Pluvium coincides 
-with Urna Mala, though they cannot be in our space 
together ; so too Moles Pluvium with Moles Mala, 
Ostrum Pluvium with Ostrum Mala. And lying towards 
us, or Y, is now that tessaract which before lay in the 
W direction from Urna, viz., Thyrsus. The order will 
therefore be the following (a star denotes the cube 
whose corner is at point of intersection of the axes, and 
the name Pluvium must be understood to follow each 
■of the names) : 

Z X Y W 
a c d 



Second Floor, j 



Second Block. 

Uncus Pallor 

Ocrea Tessera 



First Floor. 



/ Frenum Plebs 

I Crates Cura 



First Block, 

Second Floor.jO^''?'" 
I Cardo 

T- . -c■^ r*Urna Moles 

First Floor. \ „, ,,..^ 

I Thyrsus Vitta 



Bidens 
Cudo 



174 -^ New Era of Thought. 

Thus the wall of cubes in contact with that wall of the 
Mala position which contains the Urna, Moles, Ostrum, 
and Bidens Malas, is a wall composed of the Pluviums of 
Urna, Moles, Ostrum, and Bidens. The wall next to 
this, and nearer to us, is of Thyrsus, Vitta, Cardo, Cudo, 
Pluviums. The Second Block is one inch out of our 
Space, and only enters it if the Block moves Kata. 
Model 7 shows the Pluvium cube ; and each of the cubes 
of the tessaracts seen in the Pluvium position is a Pluv- 
ium. If the tessaractic Set moved Kata, we would see 
the Section between Pluvium and Tela for all but a 
minute ; and then Tela would enter our space, and the 
Tela of each tessaract would be seen. Model 12 shows 
the Section from Pluvium to Tela. Model 8 is Tela. 
Tela only lasts for a flash, as it has only the minutest 
magnitude in the unknown or Ana direction. Then, 
Frenum Pluvium takes the place of Urna Tela ; and, 
when it passes through, we see a similar section between 
Frenum Pluvium and Frenum Tela, and lastly Frenum 
Tela. Then the tessaractic Set passes out, or Kata, our 
space. A similar process takes place with every other 
tessaract, when the Set of tessaracts moves through our 
space. 

There is still one more arrangement to be learnt. If 
the line of the tessaract, which in the Mala position goes 
Ana, or W, be changed into the Z or downwards direc- 
tion, the tessaract will then appear in our space below the 
Mala position ; and the side presented to us will not be 
Mala, but that which contains the lines Dos, Cuspis, and 
Ops. This side is Model 3, and is called Lar. Under- 
neath the place which was occupied by Urna Mala, will 
come Urna Lar ; under the place of Moles Mala, Moles 
Lar ; under the place of Frenum Mala, Frenum Lar. 
The tessaract, which in the Mala position was an inch 
out of our space Ana, or W, from Urna Mala, will now 



Representation of Four -Space by Name. 175 

come into it an inch downwards, or Z, below Urna 
Mala, with its Lar presented to us ; that is, Thyrsus 
Lar will be below Urna Lar. In the whole arrange- 
ment of them written below, the highest floors are 
written first, for now they stretch downwards instead of 
upwards. The name Lar is understood after each. 

Z X Y W 
c d a 

Second Block. 

, ^, f Uncus Pallor 

Second Floor. i ^ . „., 

tOstrum Bidens 

_. _, fOcrea Tessera 

First Floor. \ n a /- j 

LCardo Cudo 

First Block. 

Frenum Plebs 



•In 



Second Floor.. ^U^^^ Moles 

Crates Cura 

Thyrsus Vitta 



First Floor. \ 



Here it is evident that what was the lower floor of 
Malas, Urna, Moles, Plebs, Frenum, now appears as if 
carried downwards instead of upwards, Lars being pre- 
sented in our space instead of Malas. This Block of 
Lars is what we see of the tessaract Set when the 
Arctos line, which in the Mala position goes up, is 
turned into the Ana, or W, direction, and the Ops line 
comes in downwards. 

The rest of the tessaracts, which consists of the cubes 
opposite to the four treated above, and of the tessaractic 
space between them, is all Ana our space. If the tessar- 
act be moved through our space — for instance, when the 
Lars are present in it — we see, taking Urna alone, first 
the section between Urna Lar and Urna Velum (Model 



176 A New Era of Thought. 

10), and then Urna Velum (Model 4), and similarly the 
sections and Velums of each tessaract in the Set. When 
the First Block has passed Kata our space, Ostrum 
Lar enters ; and the Lars of the Second Block of tessa- 
racts occupy the places just vacated by the Velums of 
the First Block. Then, as the tessaractic Set moves 
on Kata, the sections between Velums and Lars of the 
Second Block of tessaracts enter our space, and finally 
their Velums. Then the whole tessaractic Set disappears 
from our space. 

When we have learnt all these aspects and passages, 
we have experienced some of the principal features of 
this small Set of tessaracts. 



CHAPTER IX. 

FURTHER STUDY OF TESSARACTS. 

When the arrangement of a small set has been 
mastered, the different views of the whole 8i Set should 
be learnt. It is now clear to us that, in the list of the 
names of the eighty-one tessaracts given above, those 
which lie in the W direction appear in different blocks, 
while those that lie in the Z, X, or Y directions can be 
found in the same block. Therefore, from the arrange- 
ment given, which is denoted by " , or more 
^ ' ' a c a 0' 

briefly by a cdo, we can form any other arrangement. 

To confirm the meaning of the symbol a cdo for 
position, let us remember that the order of the axes 
known in our space will invariably be Z X Y, and the 
unknown direction will be stated last, thus : Z X Y W. 
Hence, if we write addc, we know that the position or 
aspect intended is that in which Arctos {a) goes Z, Ops 
(5) negative X, Dos (d) Y, and Cuspis {c) W. And such 
an arrangement can be made by shifting the nine cubes 
on the left side of the First Block of the eighty-one tes- 
saracts, and putting them into the Z X Y octant, so that 
they just touch their former position. Next to them, to 
their left, we set the nine of the left side of the Second 
Block of the 8i Set ; and next to these again, on their 
left, the nine of the left side of the Third Block. This 
Block of twenty-seven now represents Vesper Cubes, 
which have only one square identical with the Mala 

177 N 



1 78 A New Era of Thought. 

cubes of the previous blocks, from which they were 
taken. 

Similarly the Block which is one inch Ana, can be 
made by taking the nine cubes which come vertically 
in the middle of each of the Blocks in the first position, 
and arranging them in a similar manner. Lastly, the 
Block which lies two inches Ana, can be made by taking 
the right sides of nine cubes each from each of the three 
original Blocks, and arranging them so that those in the 
Second original Block go to the left of those in the First, 
and those in the Third to their left. In this manner we 
should obtain three new Blocks, which represent what 
we can see of the tessaracts, when the direction in which 
Urna, Moles, Saltus lie in the original Set, is turned W. 

The Pluvium Block we can make by taking the front 
wall of each original Block, and setting each an inch 
nearer to us, that is — Y. The far sides of these cubes 
are Moenas of Pluviums. By continuing this treatment 
of the other walls of the three original Blocks parallel to 
the front wall, we obtain two other Blocks of tessaracts. 
The three together are the tessaractic position acod, for 
in all of them Ops lies jn the — Y direction, and Dos 
has been turned W. 

The Lar position is more difficult to construct. To 
put the Lars of the Blocks in their natural position in 
our space, we must start with the original Mala Blocks, 
at least three inches above the table. The First Lar 
Block is made by taking the lowest floors of the three 
Mala Blocks, and placing them so that that of the 
Second is below that of the First, and that of the Third 
below that of the Second. The floor of cubes whose 
diagonal runs from Urna Lar to Remus Lar, will be at 
the top of the Block of Lars ; and that whose diagonal 
goes from Cervix Lar to Angusta Lar, will be at the 
bottom. The next Block of Lars will be made by 



Ftirther Study of Tessaracts. 179 

taking the middle horizontal floors of the three original 
Blocks, and placing them in a similar succession — the 
floor from Ostrum Lar to Aer Lar being at the top, that 
from Cardo Lar to Colus Lar in the middle, and Verbum 
Lar to Tabula Lar at the bottom. The Third Lar 
Block is composed of the top floor of the First Block on 
the top — that is, of Comes Lar to Tyro Lar, of Cortex 
Lar to Pluma Lar in the middle, and Axis Lar to Portio 
Lar at the bottom. 



CHAPTER X. 

CYCLICAL PROJECTIONS. 

Let us denote the original position of the cube, that 
wherein Arctos goes Z, Cuspis X, and Dos Y, by the 
expression, 

ZX Y 

a c d (") _ 

If the cube be turned round Cuspis, Dos goes Z, 
Cuspis remains unchanged, and Arctos goes Y, and we 
have the position, 

Z X Y 
d c a 

Z 

where -^ means that Dos runs in the negative direction 
d 

of the Z axis from the point where the axes intersect. 

7 Z 

We might write ^ but it is preferable to write -. 

If we next turn the cube round the line, which runs 
Y, that is, round Arctos, we obtain the position, 
ZX Y ■ 

c d a (2) 

and by means of this double turn we have put c and d 
in the places of a and c. Moreover, we have no nega- 
tive directions. This position we call simply c d a. 
If from it we turn the cube round a, which runs Y, 

Z X Y 

we get - ' and if, then, we turn it round Dos we get 

Z X Y 

, or simply d a c , This last is another position in 

180 



Cyclical Projections. 



i8l 



which all the lines are positive, and the projections, in- 
stead of lying in different quadrants, will be contained 
in one. 

The arrangement of cubes in acd v^z know. That 
'\x\ c d a\s: 



Third 
Floor. 



i Vestis 
< Scena 
(. Saltus 



Second ( Tibicen 

Floor. Kl f"^ 
(. Moles 

• Comes 
Ostrum 
. Urna 



First 
Floor. 



Oliva 

Tergum 

Sypho 

Mora 
Pallor 
Plebs 

Spicula 

Uncus 

Frenum 



Tyro 

Aer 

Remus 

Merces 

Cortis 

Hama 

Mars 

Ala 

Sector 



It will be found that learning the cubes in this position 
gives a great advantage, for thereby the axes of the cube 
become dissociated with particular directions in space. 

The d a c position gives the following arrangement: 



Remus 


Aer 


Tyro 


Hama 


Cortis 


Merces 


Sector 


Ala 


Mars 


Sypho 


Tergum 


Oliva 


Plebs 


Pallor 


Mora 


Frenum 


Uncus 


Spicula 


Saltus 


Scena 


Vestis 


Moles 


Bidens 


Tibicen 


Urna 


Ostrum 


Comes 



The side.s, which touch the vertical plane in the first 
position, are respectively, m a c d Moena, \x\ c d a Syce, 
m d a c Alvus. 

Take the shape Urna, Ostrum, Moles, Saltus, Scena, 
Sypho, Remus, Aer, Tyro. This gives vs\ a c d the 
projection : Urna Moena, Ostrum Moena, Moles Moena, 



1 82 A New Era of Thought. 

Saltus Moena, Scena Moena, Vestis Moena. (If the 
different positions of the cube are not well known, it is 
best to have a list of the names before one, but in every 
case the block should also be built, as well as the names 
used.) The same shape in the position c d a is, of course, 
expressed in the same words, but it has a different ap- 
pearance. The front face consists of the Syces of 



Saltus 


Sypho 


Remus 


Moles 


Plebs 


Hama 


Urna 


Frenum 


Sector 



And taking the shape we find we have Urna, and we 
know that Ostrum lies behind Urna, and does not come 
in ; next we have Moles, Saltus, and we know that 
Scena lies behind Saltus and does not come in ; lastly, 
we have Sypho and Remus, and we know that Aer and 
Tyro are in the Y direction from Remus, and so do not 
come in. Hence, altogether the projection will consist 
only of the Syces of Urna, Moles, Saltus, Sypho, and 
Remus. 

Next, taking the position d a c, the cubes in the front 
face have their Alvus sides against the plane, and are : 



Sector 


Ala 


Mars 


Frenum 


Uncus 


Spicula 


Urna 


Ostrum 


Comes 



And, taking the shape, we find Urna, Ostrum ; Moles 
and Saltus are hidden by Urna, Scena is behind Ostrum, 
Sypho gives Frenum, Remus gives Sector, Aer gives Ala, 
and Tyro gives Mars. All these are Alvus sides. 

Let us now take the reverse problem, and, given the 
three cyclical projections, determine the shape. Let 
the ac d projection be the Moenas of Urna, Ostrum, 
Bidens, Scena, Vestis. Let MSxt, c d a be the Syces of 
Urna, Frenum, Plebs, Sypho, and \hft d a c be the Alvus 
of Urna, Frenum, Uncus, Spicula. Now, from a c d viC 



Cyclical Projections. i8 



p 



have Urna, Frenum, Sector, Ostrum, Uncus, Ala, Bidens, 
Pallor, Cortis, Scena, Tergum, Aer, Vestis, Oliva, Tyro. 
From c d a we have Urna, Ostrum, Comes, Frenum, 
Uncus, Spicula, Plebs, Pallor, Mora, Sypho, Tergum, 
Oliva. In order to see how these will modify each 
other, let us consider the a c d solution as if it were a 
set of cubes in the c d a arrangement. Here, those that 
go in the Arctos direction, go away from the plane of 
projection, and must be represented by the Syce of the 
cube in contact with the plane. Looking at the a c d 
solution we write down (keeping those together which go 
away from the plane of projection) : Urna and Ostrum, 
Frenum and Uncus, Sector and Ala, Bidens, Pallor, 
Cortis, Scena and Vestis, Tergum and Oliva, Aer and 
Tyro. Here we see that the whole c d a face is filled up 
in the projection, as far as this solution is concerned. 
But in the c d a solution we have only Syces of Urna, 
Frenum, Plebs, Sypho. These Syces only indicate the 
presence of a certain number of the cubes stated above 
as possible from the Moena projection, and those are 
Urna, Ostrum, Frenum, Uncus, Pallor, Tergum, Oliva. 
This is the result of a comparison of the Moena pro- 
jection with the Syce projection. Now, writing these 
last named as they come in the d ac projection, . we 
have Urna, Ostrum, Frenum, Uncus and Pallor and 
Tergum, Oliva. And of these Ostrum Alvus is wanting 
in the d ac projection as given above. Hence Ostrum 
will .be wanting in the final shape, and we. have as the 
final solution : Urna, Frenum, Uncus, Pallor, Tergum, 
Oliva. 



CHAPTER XI. 

A TESSARACTIC FIGURE AND ITS PROJECTIONS. 

We will now consider a fourth-dimensional shape com- 
posed of tessaracts, and the manner in which we can 
obtain a conception of it. The operation is precisely 
analogous to that described in chapter VI., by which a 
plane being could obtain a conception of solid shapes. 
It is only a little more difficult in that we have to deal 
with one dimension or direction more, and can only do 
so symbolically. 

We will assume the shape to consist of a certain 
number of the 8i tessaracts, whose names we have 
given on p. i68. Let it consist of the thirteen tessaracts r 
Urna, Moles, Plebs, Frenum, Pallor, Tessera, Cudo, Vitta, 
Cura, Penates, Polus, Orcus, Lacerta. 

Firstly, we will consider what appearances or projec- 
tions these tessaracts will present to us according as the 
tessaractic set touches our space with its {a) Mala cubes, 
(b) Vesper cubes, {c) Pluvium cubes, or {d) Lar cubes. 
Secondly, we will treat the converse question, how the 
.shape can be determined when the projections in each 
of those views are given. 

Let us build up in cubes the four different arrange- 
ments of the tessaracts according as they enter our space 
on their Mala, Vesper, Pluvium or Lar sides. They can 
only be printed by symbolizing two of the directions. 
In the following tabulations the directions Y, X will at 



A Tessaractic Figure and its Projections. 185 

once be understood. The direction Z (expressed by the 
wavy line) indicates that the floors of nine, each printed 
nearer the top of the page, lie above those printed nearer 
the bottom of it. The direction W is indicated by the 
dotted line, which shows that the floors of nine lying to 
the left or right are in the W direction (Ana) or the - W 
direction (Kata) from those which lie to the right or 
left. For instance, in the arrangement of the tessaracts, 
as Malas (Table A) the tessaract Tessara, which is 
exactly in the middle of the eighty-one tessaracts has 



Domitor on its right side or 


in the 


X direction. 


Ocrea on its left „ 


>> 


-X 


»» 


Glans away from us „ 


>j 


Y 


^ J) 


Cudo nearer to us „ 


»» 


-Y 


)1 


Sacerdos above it „ 


ti 


Z 


J> 


Cura below it „ 




-Z 


J) 


Lacerta in the Ana or 




W 


J» 


Pallor in the Kata or 




-W 


)* 



Similarly Cervix lies in the Ana or W direction from 
Urna, with Thyrsus between them. And to take one 
more instance, a journey from Saltus to Arcus would 
be made by travelling Y to Remus, thence - X to Sector, 
thence Z to Mars, and finally W to Arcus. A line from 
Saltus to Arcus is therefore a diagonal of the set of 
81 tessaracts, because the full length of its side has 
been traversed in each of the four directions to reach 
one from the other, i.e. Saltus to Remus, Remus to 
Sector, Sector to Mars, Mars to Arcus. 

The relation between the four different arrangements 
shown in the tables A, B, C, and D, will be understood 
from what has been said in chapter VIII. about a small 
set of sixteen tessaracts. A glance at the lines, which 
indicate the directions in each, will show the changes 



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iQO A New Era of Thought. 

effected by turning the tessaracts from the Mala presen- 
tation. 

In the Vesper presentation : 
The tessaracts — 

{e.g. Urna, Ostrum, Comes), which ran Z still run Z. 
{e.g. Urna, Frenum, Sector), „ Y „ Y. 

{e.g. Urna, Moles, Saltus), „ X now run W. 

{e.g. Urna, Thyrsus, Cervix), „ W „ - X. 

In the Pluvium presentation : 
The tessaracts — 

{e.g. Urna, Ostrum, Comes), which ran Z still run Z. 
{e.g. Urna, Moles, Saltus), „ X „ X. 

{e.g. Urna, Frenum, Sector), „ Y now run W. 

{e.g. Urna, Thyrsus, Cervix), „ W „ - Y. 

In the Lar presentation : 
The tessaracts — 

{e.g. Urna, Moles, Saltus), which ran X still run X. 

{e.g. Urna, Frenum, Sector), „ Y „ Y. 

{e.g. Urna, Ostrum, Comes), „ Z now run W. 

{e.g. Urna, Thyrsus, Cervix), „ W „ - Z. 

This relation was already treated in chapter IX., but 
it is well to have it very clear for our present purpose. 
For it is the apparent change of the relative positions 
of the tessaracts in each presentation, which enables us 
to determine any body of them. 

In considering the projections, we always suppose our- 
selves to be situated Ana or W towards the tessaracts, 
and any movement to be Kata or -W through our 
space. For instance, in the Mala presentation we have 
first in our space the Malas of that block of tessaracts, 
which is the last in the - W direction. Thus, the Mala 
projection of any given tessaract of the set is that Mala 



A Tessaractic Figure and its Projections. 191 

in the extreme - W block, whose place its (the given 
tessaract's) Mala would occupy, if the tessaractic set 
moved Kata until the given tessaract reached our space. 
Or, in other words, if all the tessaracts were transparent 
except those which constitute the body under considera- 
tion, and if a light shone through Four-space from the 
Ana ( W) side to the Kata ( - W) side, there would be 
darkness in each of those Malas, which would be occu- 
pied by the Mala of any opaque tessaract, if the tes- 
saractic set moved Kata. 

Let us look at the set of 81 tessaracts we have built 
up in the Mala arrangements, and trace the projections 
in the extreme - W block of the thirteen of our shape. 
The latter are printed in italics in Table A, and their 
projections are marked J. 

Thus the cube Uncus Mala is the projection of the 
tessaract Orcus, Pallor Mala of Pallor and Tessera and 
Tacerta, Bidens Mala of Cudo, Frenum Mala of Frenum 
and Polus, Plebs Mala of Plebs and Cura and Penates, 
Moles Mala of Moles and Vitta, Urna Mala of Urna. 

Similarly, we can trace the Vesper projections (Table 
B). Orcus Vesper is the projection of the tessaracts 
Orcus and Lacerta, Ocrea Vesper of Tessera, Uncus 
Vesper of Pallor, Cardo Vesper of Cudo, Polus Vesper 
of Polus and Penates, Crates Vesper of Cura, Frenum 
Vesper of Frenum and Plebs, Urna Vesper of Urna and 
Moles, Thyrsus Vesper of Vitta. Next in the Pluvium 
presentation (Table C) we find that Bidens Pluvium is 
the projection of the tessafact Pallor, Cudo Pluvium of 
Cudo and Tessera, Luctus Pluvium of Lacerta, Verbum 
Pluvium of Orcus, Urna Pluvium of Urna and Frenum, 
Moles Pluvium of Moles and Plebs, Vitta Pluvium of 
Vitta and Cura, Securis Pluvium of Penates, Cervix 
Pluvium of Polus. Lastly, in the Lar presentation 
(Table D) we observe that Frenum Lar is the projection 



192 A New Era of Thought. 

of Frenum, Plebs Lar of Plebs and Pallor, Moles Lar 
of Moles, Urna Lar of Urna, Cura Lar of Cura and 
Tessara, Vitta Lar of Vitta and Cudo, Penates Lar of 
Penates and Lacerta, Polur Lar of Polus and Orcus. 

Secondly, we will treat the converse problem, how to 
determine the shape when the projections in each pre- 
sentation are given. Looking back at the list just given 
above, let us write down in each presentation the pro- 
jections only. 

Mala projections : 

Uncus, Pallor, Bidens, Frenum, Plebs, Moles, Urna. 
Vesper projections : 

Orcus, Ocrea, Uncus, Cardo, Polus, Crates, Frenum, 
Urna, Thyrsus. 
Pluvium projections : 

Bidens, Cudo, Luctus, Verbum, Urna, Moles, Vitta, 
Securis, Cervix. 
Lar projections : 

Frenum, Plebs, Moles, Urna, Cura, Vitta, Polus, 
Penates. 

Now let us determine the shape indicated by these 
projections. In now using the same tables we must not 
notice the italics, as the shape is supposed to be un- 
known. It is assumed that the reader is building the 
problem in cubes. From the Mala projections we might 
infer the presence of all or any of the tessaracts written 
in the brackets in the following list of the Mala presen- 
tation. 

(Uncus, Ocrea, Orcus) ; (Pallor, Tessera, Lacerta) ; 

(Bidens, Cudo, Luctus) ; (Frenum, Crates, Polus) ; 

(Plebs, Cura, Penates) ; (Moles, Vitta, Securis) ; 

(Urna, Thyrsus, Cervix). 

Let us suppose them all to be present in our shape, 



A TessaracHc Figure and its Projections. 193 

and observe what their appearance would be in the 
Vesper presentation. We mark them all with an asterisk 
in Table B. In addition to those already marked we 
must mark (f) Verbum, Cardo, Ostrum, and then we 
see all the Vesper projections, which would be formed 
by all the tessaracts possible from the Mala projections. 
Let us compare these Vesper projections, viz. Orcus, 
Ocrea, Uncus, Verbum, Cardo, Ostrum, Polus, Crates, 
Frenum, Cervix, Thyrsus, Urna, with the given Vesper 
projections. We see at once that Verbum, Ostrum, and 
Cervix are absent. Therefore, we may conclude that 
all the tessaracts, which would be implied as possible by 
their presence, are absent, and of the Mala possibili- 
ties may exclude the tessaracts Bidens, Luctus, Securis, 
and Cervix itself Thus, of the 21 tessaracts possible 
in the Mala view, there remain only 17 possible, both 
in the Mala and Vesper views, viz. Uncus, Ocrea, 
Orcus, Pallor, Tessera, Lacerta, Cudo, Frenum, Crates, 
Polus, Plebs, Cura, Penates, Moles, Vitta, Urna, Thyrsus. 
This we call the Mala- Vesper solution. 

Next let us take the Pluvium presentation. We again 
mark with an asterisk in Table C the possibilities in- 
ferred from the Mala-Vesper solution, and take the 
projections those possibilities would produce. The ad- 
ditional projections are again marked (t). There are 
twelve Pluvium projections altogether, viz. Bidens, Os- 
trum, Cudo, Cardo, Luctus, Verbum, Urna, Moles, Vitta, 
Thyrsus, Securis, Cervix. Again we compare these with 
the given Pluvium projections, and find three are absent, 
viz. Ostrum, Cardo, Thyrsus. Hence the tessaracts 
implied by Ostrum and Cardo and Thyrsus cannot be 
in our shape, viz. Uncus, Ocrea, Crates, nor Thyrsus 
itself Excluding these four from the seventeen possi- 
bilities of the Mala-Vesper solution we have left the 
thirteen tessaracts : Orcus, Pallor, Tessera, Lacerta, Cudo, 

O 



194 -^ New Era of Thought. 

Frenum, Polus, Plebs, Curd, Penates, Moles, Vitta, Urna. 
This we call the Mala-Vesper-Pluvium solution. 

Lastly, we have to consider whether these thirteen 
tessaracts are consistent with the given Lar projections. 
We mark them again on Table D with an asterisk, and 
we find that the projections are exactly those given, viz. 
Frenum, Plebs, Moles, Urna, Cura, Vitta, Polus, Penates. 
Therefore, we have not to exclude any of the thirteen, 
and can infer that they constitute the shape, which 
produces the four different given views or projections. 

In fine, any shape in space consists of the possibilities 
common to the projections of its parts upon the boun- 
daries of that space, whatever be the number of its 
dimensions. Hence the simple rule for the determina- 
tion of the shape would be to write down all the possi- 
bilities of the sets of projections, and then cancel all 
those possibilities which are not common to all. But 
the process adopted above is much preferable, as through 
it we may realize the gradual delimitation of the shape 
view by view. For once more we must remind ourselves 
that our great object is, not to arrive at results by 
symbolical operations, but to realize those results piece 
by piece through realized processes. 



APPENDICES. 






u 



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A New Era of Thought. 







APPENDIX B. 






The following list of names is 


; used to denote cubic spaces. 


makes a cubic block of six floor: 


i, the highest being the sixth. 


^ Fons ' 


Plectrum 


Vulnus 


Arena 


Mensa 


Terminus 


1 Testa 


Plausus 


Uva 


CoUis 


Coma 


Nebula 


kI Copia 


Cornu 


Solum 


Munus 


Rixum 


Vitrum 


■S Ars 


Fervor 


Thyma 


Colubra 


Seges 


Cor 


^ Lupus 
Thalamus 


Classis 


Modus 


Flamma 


Mens 


Incola 


Hasta 


Calamus 


Crinis 


Auriga 


Vallum 


Linteum 


Pinnis 


Puppis 


Nuptia 


Aegis 


Cithara 


§ Triumphus Curris 


Lux 


Portus 


Latus 


Funis 


Ei^ Regnum 


Fascis 


Bellum 


Capellus 


Arbor 


Custos 


«e Sagitta 


Puer 


Stella 


Saxum 


Humor 


Pontus 


1^ Nomen 


Imago 


Lapsus 


Quercus 


Mundus 


Proelium 


Palaestra 


Nuncius 


Bos 


Pharetra 


Pumex 


Tibia 


v; Lignum 


Focus 


Omus 


Lucrum 


Alea 


Vox 


1 Caterva 


Facias 


Onus 


Silva 


Gelu 


Flumen 


"*- Tellus 


Sol 


Os 


Arma 


Brachium 


1 Jaculum 


'^ Merum 


Signum 


Umbra 


Tempus 


Corona 


Socius 


1 Moena 


Opus 


Honor 


Campus 


Rivus 


Imber 


'*< Victor 


Equus 


Miles 


Cursus 


Lyra 


Tunica 


^- Haedus 


Taberna 


Turris 


Nox 


Domus 


Vinum 


§ Pruinus 


Chorus • 


Luna 


Flos 


Lucus 


Agna 


i*^ Fulmen 


Hiems 


Ver 


Carina 


Arator 


Pratum 


[5 Oculus 


Ignis 


Aether 


Cohors 


Penna 


Labor 


g Aes 
Princeps 


Pectus 


Pelagus 


Notus 


Fretum 


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Dux 


Ventus 


Navis 


Finis 


Robur 


v: Vultus 


Hostis 


Figura 


Ales 


Coelum 


Aura 


^ Humerus 


Augur 


Ludus 


Clamor 


Galea 


Pes 


J^ Civis 


Ferrum 


Pugna 


Res 


Carmen 


Nubes 


R Litus 


Unda 


Rex 


Templum 


Ripa 


Amnis 


^ Pannus 


Ulmus 


Sedes 


Columba 


Aequor 


Dama 


^ Dexter 


Urbs 


Gens 


Monstrum Pecus 


Mons 


Nemus 


Sidus 


Vertex 


Nix 


Grando 


Arx 


§ Venator 


Cei-va 


Aper 


Plagua 


Hedera 


Frons 


E^ Membrum Aqua 


Caput 


Castrum 


Lituus 


Tuba 


•;» Fluctus 


Rus 


Ratis 


Amphora 


Pars 


Dies 


i^ Turba 
Decus 


Ager 


Trabs 


Myrtus 


Fibra 


Nauta 


Pulvis 


Meta 


Rota 


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<; u Ph pq 




Rub 
Han 
Pleb 
Mol 






K < W < 





43 tn 




,n 3 <U 


X 


Mar; 
Spic 
Com 


tri 


|Jh 



o ^ c -K 

p^ <: t3 o 



tn ^ S 

S S^ 5i S 
Ph W [i, P 



C tn 

)-i tn ff 3 

lU 3 3 a 

B c > S 

< W< P^ 



3 .3 3 tn 

<i O J g 



iS tn t^ 

.2 3 2 o 

>-, tu m ^ 

<l Ps h H 



E >- 

3 g 

o S rt >< 

lU 3 .t! (ti 

H P^ > Ph 



t4 
U 

o 

m 

p 

o 
o 



W < Pi < 



-SI'S g 



B 

_3 

art cj f-H 
«, c ^ &^ 

S E 3 >;>JS 
^ O Pi III h 



R 



o 



■ „ o " 

1-1 > OT S 



§ ^ o o rt 
jS Ph U Q § 

8 



2 t« 



Pil O H U 



I 

^ § < J w 



.2 3 d d 

C " >-I S 

3 d 3 .-t! 

U J U > 



X u 
tn j; o Hi 
3 S'^ « 

3 3 0. 

<1 < h w 



cj O Ji 

<; Pi o 



fc C tTi O 

o S lu -a 
< P< O U 



n! 13 „ 3 

.S K S K 

bo C ts >-. 

P2l < U H 



202 



A New Era of Thought. 



APPENDIX D. 

The following list gives the colours, and the various uses for 
them. They have already been* used in the foregoing pages to 
distinguish the various regions of the Tessaract, and the different 
individual cubes or Tessaracts in a block. The other use suggested 
in the last column of the list has not been discussed ; but it is be- 
lieved that it may afford great aid to the mind in amassing, 
handling, and retaining the quantities of formulae requisite in 
scientific training and work. 



Colour. 

Black 

White 

Vermilion 

Orange 

Light-yellow 

Bright-green 

Bright-blue 

Light-grey 

Indian-red 

Yellow-ochre 

Buff 

Wood 

Brown-green 

Sage-green 

Reddish 

Chocolate 

French-grey 

Brown 

Dark-slate 

Dun 

Orange-vermilion 

Stone 

Quaker-green 

Leaden 

Dull-green 

Indigo 

Dull-blue 

Dark-purple 

Pale-pink 

Dark-blue 

Earthen 

Blue 

Terracotta 

Oak 

Yellow 



Region of Tessaract in 


Tessaract. 


81 Set. 


Syce 


Plebs 


Mel 


Mora 


Alvus 


Uncus 


Cuspis 


Moles 


Murex 


Cortis 


Lappa 


Penates 


Iter 


Oliva 


Lares 


Tigris 


Crux 


Orcus 


Sal 


Testudo 


Cista 


Sector 


Tessaraci 


: Tessara 


Tholus 


Troja 


Margo 


Lacerta 


Callis 


Tibicen 


Velum 


Sacerdos 


Far 


Scena 


Arctos 


Ostrum 


Daps 


Aer 


Portica 


Clipeus 


in Talus 


Portio 


Ops 


Thyrsus 


Felis 


Axis 


Semita 


Merces 


Mappa 


Vulcan 


Lixa 


Postis 


Pagus 


Verbnm 


Mensura 


Nepos 


. Vena 


Tabula 


Moena 


Bidens 


Mugil 


Angusta 


Dos 


Frenum 


Crus 


Remus 


Idus 


Domitor 


Pagina 


Cardo 



Symbol. 



o 
I 

3 
3 
4 
S 
6 

7 



-f (plus) 

— (minus) 

± (plus or minus) 

X (multiplied by) 

-7- (divided by) 

^ (equal to) 

iXz (not equal to) 

> (greater than) 

< (less than) 

: I signs 

: : ( of proportion 

■ (decimal point) 

l_ (factorial) 

II (parallel) 

14 (not parallel) 

~ (90°) (at right angles) 

log. base 10 

sin. (sine) 

cos. (cosine) 

tan. (tangent) 

00 (infinity) 

a 

b 

c 

d 



Appendix. 



203 



Colour, 



Region of Tessaract in 
Tessaract. 81 Set. 



Symbol. 



Green 


Bucina 


Ala 


e 


Rose 


OUa 


Limen 


f 


Emerald 


Orsa 


Ara 


g 


Red 


Olus 


Mars 


h 


Sea-green 


Libera 


Pluma 


i 


Salmon 


Tela 


Glans 


j 


Pale-yellow 


Livor 


Ovis 


k 


Purple-brown 


Opex 


Polus 


\ 


Deep-crimson 


Camoena 


Pilum 


m 


Blue-green 


Proes 


Tergum 


n 


Light-brown 


Lua 


Crates 





Deep-blue 


Lama 


Tyro 


P 


Brick-red 


Lar 


Cura 


q 


Magenta 


Offex 


Arvus 


r 


Green-grey 


Cadus 


Hama 


s 


Light-red 


Croeta 


Praeda 


t 


Azure 


Lotus 


Vitta 


u 


Pale-green 


Vesper 


Ocrea 


V 


Blue-tint 


Panax 


Telum 


W 


Yellow-green 


Pactum 


Malleus 


X 


Deep-green 


Mango 


Vomer 


y 


Light-green 


Lis 


Agmen 


z 


Light-blue 


Ilex 


Comes 


a 


Crimson 


Bolus 


Sypho 


3 


Ochre 


Limbus 


Mica 


V 


Purple 


Solia 


Arcus 


t 


Leaf-green 


Luca 


Securis 


E 


Turquoise 


Ancilla 


Vinculum 


i 


Dark-grey 


Orca 


Colus 


V 


Fawn 


Nugffi 


Saltus 


e 


Smoke 


Limus 


Sceptrum 


I 


Light-buff 


Mala 


Pallor 


K 


Dull-purple 


Sors 


Vestis- 


X 


Rich-i«d 


Lucta 


Cortex 


t^ 


Green-blue 


Pator 


Flagellum 


V 


Burnt-sienna 


Silex 


Luctus 


i 


Sea-blue 


Lorica 


Lacus 





Peacock-blue 


Passer 


Aries 


IT 


Deep-brown 


Meatus 


Hydra 


P 


Dark-pink 


Onager 


Anguis 


<T 


Dark 


Lensa 


Laurus 


T 


Dark-stone 


Pluvium 


Cudo 


V 


Silver 


Spira 


Cervix 





Gold 


Corvus 


Urna 


X 


Deep-yellow 


Via 


Spicula 


^ 


Dark-green 


Calor 


Segmen 


CO 



204 A New Era of Thought. 



APPENDIX E. 
A Theorem in Four-space. 

If a pyramid on a triangular base be cut by a plane which passes 
through the three sides of the pyramid in such manner that the sides 
of the sectional triangle are not parallel to the corresponding sides 
of the triangle of the base ; then the sides of these two triangles, il 
produced in pairs, will meet in three points which are in a straight 
line, namely, the line of intersection of the sectional plane and the 
plane of the base. 

Let A B C D be a pyramid on a triangular base ABC, and let 
a b c be a section such that A B, B C, A C, are respectively not 
parallel to a b, be, a c. It must be understood that a is a point 
on A D, b is a point on B D, and c a point on C D. Let, A B and 
a b, produced, meet in m. B C and b c, produced, meet in n ; and 
A C and a c, produced, meet in o. These three points, m, n, o, 
are in the line of intersection of the two planes ABC and a b c. 

Now, let the line a b be projected on to the plane of the base, by 
drawing lines from a and b at right angles to the base, and meeting 
it in a' b' ; the line a' b', produced, will meet A B produced in m. 
If the lines 1) c and a c be projected in the same way on to the 
base, to the points b' c' and a' c' ; then B C and b' c' produced, 
will meet in n, and A C and a' c' produced, will meet in o. The 
two triangles ABC and a' b' c' are such, that the lines joining 
A to a', B to b', and C to c', will, if produced, meet in a point, 
namely, the point on the base ABC which is the projection of D. 
Any two triangles which fulfil this condition are the possible base 
and projection of the section of a pyramid ; therefore the sides of 
such triangles, if produced in pairs, will meet (if they are not 
parallel) in three points which lie in one straight line. 

A four-dimensional pyramid may be defined as a figure bounded 
by a polyhedron of any number of sides, and the same number of 
pyramids whose bases are the sides of the polyhedron, and whose 
apices meet in a point not in the space of the base. 

If a four-dimensional pyramid on a tetrahedral base be cut by a 
space which passes through the four sides of the pyramid in such 
a way that the sides of the sectional figure be not parallel to the 
sides of the base ; then the sides of these two tetrahedra, if produced 
in pairs, will meet in lines which all lie in one plane, namely, the 
plane of intersection of the space of the base and the space of the 
section. 



Appendix. 205 

If now the sectional tetrahedron be projected on to the base (by 
drawing lines from each point of the section to the base at right 
angles to it), there will be two tetrahedra fulfilling the condition 
that the hne joining the angles of the one to the angles of the 
other will, if produced, meet in a point, which point is the projec- 
tion of the apex of the four-dimensional pyramid. 

Any two tetrahedra which fulfil this condition, are the possible 
base and projection of a section of a four-dimensional pyramid. 
Therefore, in any two such tetrahedra, where the sides of the one 
a:re not parallel to the sides of the other, the sides, if produced in 
pairs (one side of the one with one side of the other), will meet in 
four straight lines which are all in one plane. 



APPENDIX F. 

Exercises on Shapes of Three Dimensions. 
The names used are those given in Appendix B. 

Find the shapes from the following projections : 

1. Syce projections : Ratis, Caput, Castrum, Plagua. 
Alvus projections : Merum, Oculus, Fulmen, Pruinus. 
Moena projections : Miles, Ventus, Navis. 

2. Syce : Dies, Tuba, Lituus, Frons. 

Alvus : Sagittk, Regnum, Tellus, Fulmen, Pruinus. 
Moena : Tibia, Tunica, Robur, Finis. 

3. Syce : Nemus, Sidus, Vertex, Nix, Cerva. 

Alvus : Lignum, Haedus, Vultus, Nemus, Humerus. 
Moena : Dexter, Princeps, Equus, Dux, Urbs, Pullis, Gens, 
Monstrum, Miles. 

4. Syce : Amphora, Castrum, Myrtus, Rota, Palma, Meta, Trabs, 

Ratis. 

Alvus:, Dexter, Princeps, Moena, Aes, Merum, Oculus, Littus, 
Civis, Fulmen. 

Moena : Gens, Ventus, Navis, Finis, Monstrum, Cursus. 
;. Syce : . Castrum, Plagua, Nix, Vertex, Aper, Caput, Cerva, 
Venator. 

Alvus : Triumphus, Tellus, Caterva, Lignum, Haedus, Pruinus, 
Fulmen, Civis, Humerus, Vultus. 

Moena : Pharetra, Cursus, Miles, Equus, Dux, Navis, Mon- 
strum, Gens, Urbs, Dexter. 



2o6 A New Era of Thought. 

Answers.- 
The shapes are : 

1. Umbra, Aether, Ver, Carina, Flos. 

2. Pontus, Custos, Jaculum, Pratum, Arator, AgnaT. 

3. Focus, Omus, Haedus, Tabema, Vultus, Hostis, Figura, Ales, 

, Sidus, Augur. 

4. Tempus, Campus, Finis, Navis, Ventus, Pelagus, Notus, Cohors, 

Aether, Carina, Res, Templum, Rex, Gens, Monstrum. 

5. Portus, Arma, Sylva, Lucrum, Omus, Onus, Os, Facies, Chorus, 

Carina, Flos, Nox, Ales, Clamor, Res, Pugna, Ludus, 
Figura, Augur, Humerus. 

Further Exercises in Shapes of Three Dimensions. 

The Names used are those given in Appendix C ; and this set 
of exercises forms a preparation for their use in space of four 
dimensions. All are in the 27 Block (Uma to Syrma). 

1. Syce : Moles, Frenum, Plebs, Sypho. 

Alvus : Urna, Frenum, Uncus, Spicula, Comes. 
Moena : Moles, Bidens, Tibicen, Comes, Saltus. 

2. Syce : Urna, Moles, Plebs, Hama, Remus. 
Alvus : Urna, Frenum, Sector, Ala, Mars. 
Moena : Uma, Moles, Saltus, Bidens, Tibicen. 

3. Syce : Moles, Plebs, Hama, Remus. 

Alvus : Uma, Ostrum, Comes, Spicula, Frenum, Sector. 
Moena : Moles, Saltus, Bidens, Tibicen. 

4. Syce : Frenum, Plebs, Sypho, Moles, Hama. 
Alvus : Urna, Frenum, Uncus, Sector, Spicula. 
Moena : Uma, Moles, Saltus, Scena, Vestis. 

5. Syce : Urna, Moles, Plebs, Hama, Remus, Sector. 

Alvus : Urna, Frenum, Sector, Uncus, Spicula, Comes, Mars. 
Moena : Urna, Moles, Saltus, Bidens, Tibicen, Comes. 

6. Syce : Uma, Moles, Saltus, Sypho, Remus, Hama, Sector. 
Alvus :~ Comes, Ostrum, Uncus, Spicula, Mars, Ala, Sector. 
Moena : Urna, Moles, Saltus, Scena, Vestis, Tibicen, Comes, 

Ostrum. 

7. Syce : Sypho, Saltus, Moles, Urna, Frenum, Sector. 
Alvus : Urna, Frenum, Uncus, Spicula, Mars. 
Moena ; Saltus, Moles, Urna, Ostrum, Comes. 

8. Syce : Moles, Plebs, Hama, Sector. 

Alvus : Ostrum, Frenum, Uncus, Spicula, Mars, Ala. 
Moena : Moles, Bidens, Tibicen, Ostrum. 



Appendix. 207 

9. Syce : Moles, Saltus, Sypho, Plebs, Frenum, Sector. 
Alvus : Ostrum, Comes, Spicula, Mars, Ala. 
Moena : Ostrum, Comes, Tibicen, Bidens, Scena, Vestis. 

10. Syce : Urna, Moles, Saltus, Sypho, Remus, Sector, Frenum. 
Alvus : Urna, Ostrum, Comes, Spicula, Mars, Ala, Sector. 
Moena : Urna, Ostrum, Comes, Tibicen, Vestis, Scena, Saltus. 

11. Syce : Frenum, Plebs, Sypho, Hama. 
Alvus : Frenum, Sector, Ala, Mars, Spicula. 
Moena : Urna, Moles, Saltus, Bidens, Tibicen. 

Answers. 
The shapes are : 

1. Moles, Plebs, Sypho, Pallor, Mora, Tibicen, Spicula. 

2. Urna, Moles, Plebs, Hama, Cortis, Merces, Remus. 

3. Moles, Bidens, Tibicen, Mora, Plebs, Hama, Remus. 

4. Frenum, Plebs, Sypho, Tergum, Oliva, Moles, Hama. 

5. Urna, Moles, Plebs, Hama, Remus, Pallor, Mora, Tibicen, 

Mars, Merces, Comes, Sector. 

6. Ostrum, Comes, Tibicen, Vestis, Scena, Tergum, Oliva, Tyro, 

Aer, Remus, Hama, Sector, Merces, Mars, Ala. 

7. Sypho, Saltus, Moles, Urna, Frenum, Uncus, Spicula, Mars. 

8. Plebs, Pallor, Mora, Bidens, Merces, Cortis, Ala. 

9. Bidens, Tibicen, Vestis, Scena, Oliva, Mora, Spicula, Mars, 

Ala. 

10. Urna, Ostrum, Comes, Spicula, Mars, Tibicen, Vestis, Oliva, 

Tyro, Aer, Remus, Sector, Ala, Saltus, Scena. 

11. Frenum, Plebs, Sypho, Hama, Cortis, Merces, Mora. 



APPENDIX G. 

Exercises on Shapes of Four Dimensions. 

The Names used are those given in Appendix C. The first six 
exercises are in the 81 Set, and the rest in the 256 Set. 

I. Mala projection : Urna, Moles, Plebs, Pallor, Cortis, Merces. 
Lar projection : Urna, Moles, Plebs, Cura, Penates, Nepos. 
Pluvium projection : Urna, Moles, Vitta, Cudo, Luctus, Troja. 
Vesper projection : Urna, Frenum, Crates, Ocrea, Orcus, Postis, 
Arcus. 



2o8 A New Era of Thought. 

2. Mala : Urna, Frenutn, Uncus, Pallor, Cortis, Aer. 

Lar : Urna, Frenum, Crates, Cura, Lacus, Arvus, Angusta. 
Pluvium : Urna, Thyrsus, Cardo, Cudo, Malleus, Anguis. 
Vesper : Urna, Frenum, Crates, Ocrea, Pilum, Postis. 

3. Mala : Comes, Tibicen, Mora, Pallor. 
Lar ; Urna, Moles, Vjtta, Cura, Penates. 
Pluvium: Comes, Tibicen, Mica, Troja, Luctus. 
Vesper ; Comes, Cortex, Praeda, Laurus, Orcus. 

4. Mala : Vestis, Oliva, Tyro. 

Lar : Saltus, Sypho, Remus, Arvus, Angusta. 
Pluvium : Vestis, Flagellum, Aries. 
Vesper : Comes, Spicula, Mars, Ara, Areas. 

5. Mala : Mars, Merces, Tyro, Aer, Tergum, Pallor, Plebs. 

Lar : Sector, Hama, Lacus, Nepos, Angusta, Vulcan, Penates. 
Pluvium : Comes, Tibicen, Mica, Troja, Aries, Anguis, Luctus, 

Securis. 
Vesper : Mars, Ara, Arcus, Postis, Orcus, Polus. 

6. Mala : Pallor, Mora, Oliva, Tyro, Merces, Mars, Spicula, 

Comes, Tibicen, Vestis. 
Lar : Plebs, Cura, Penates, Vulcan, Angusta, Nepos, Telum, 

Polus, Cervix, Securis, Vinculum. 
Pluvium : Bidens, Cudo, Luctus, Troja, Axis, Aries. 
Vesper : Uncus, Ocrea, Orcus, Laurus, Arcus, Axis. 

7. Mala : Hospes, Tribus, Fragor, Aer, Tyro, Mora, Oliva. 

Lar : Hospes, Tectum, Rumor, Arvus, Angusta, Cera, Api^, 

Lapis. 
Pluvium : Acus, Torus, Malleus, Flagellum, Thorax, Aries, 

Aestas, Capella. 
Vesper : Pardus, Rostrum, Ardor, Pilum, Ara, Arcus, Aestus, 

Septum. 

8. Mala : Pallor, Tergum, Aer, Tyro, Cortis, Syrma, Ursa, Fama, 

Naxos, Erisma. 
Lar : Plebs, Cura, Limen, Vulcan, Angusta, Nepos, Cera, 

Papaver, Pignus, Messor. 
Pluvium : Bidens, Cudo, Malleus, Anguis, Aries, Luctus, Capella, 

Rheda, Rapina. 
Vesper : Uncus, Ocrea, Orcus, Postis, Arcus, Aestus, Cussis, 

Dolium, Alexis. 

9. Mala: Fama, Conjux, Reus, Torus, Acus, Myrrha, Sypho, 

Plebs, Pallor, Mora, Oliva, Alpis, Acies, Hircus. 
Lar : Missale, Fortuna, Vita, Pax, Furor, Ira, Vulcan, Penates, 
Lapis, Apis, Cera, Pignus. ' 



Appendix. 209 

Pluvium : Torus, Plenum, Pax, Thorax, Dolus, Furor, Vinculum, 
Securis, Clavis, Gurges, Aestas, Capella, Corbis. 

Vesper : Uncus, Spicula, Mars, Ocrea, Cardo, Thyrsus, Cervix, 
Verbum, Orcus, Polus, Spes, Senex, Septum, Porrum, 
Cussis, Dolium. 

Answers. 
The shapes are : 

1. Urna, Moles, Plebs, Cura, Tessara, Lacerta, Clipeus, Ovis. 

2. Urna, Frenum, Crates, Ocrea, Tessara, Glans, Colus, Tabula. 

3. Comes, Tibicen, Mica, Sacerdos, Tigris, Lacerta. 

4. Vestis, Oliva, Tyro, Pluma, Portio. 

S- Mars, Merces, Vomer, Ovis, Portio, Tabula, Testudo, Lacerta, 
Penates. 

6. Pallor, Tessara, Lacerta, Tigris, Segmen, Portio, Ovis, Arcus, 

Laurus, Axis, Troja, Ari«s. 

7. Hospes, Tribus, Arista, Pellis, Colus, Pluma, Portio, Calathus, 

Turtur, Sepes. 

8. Pallor, Tessara, Domitor, Testudo, Tabula, Clipeus, Portio, 

Calathus, Nux, Lectrum, Corymbus, Circaea, Cordax. 

9. Fama, Conjux, Reus, Fera, Thorax, Pax, Furor, Dolus, Scala, 

Ira, Vulcan, Penates, Lapis, Palus, Sepes, Turtur, Diota, 
Drachma, Python. 



APPENDIX H. 
Sections of Cube and Tessaract. 

There are three kinds of sections of a cube. 

1. The sectional plane, which is in all cases supposed to be 
infinite, can be taken parallel to two of the opposite faces of the 
cube ; that is, parallel to two of the lines meeting in Corvus, and 
cutting the third. 

2. The sectional plane can be taken parallel to one of the lines 
meeting in Corvus and cutting the other two, or one or both of 
them produced. 

3. The sectional plane can be taken cutting all three lines, or 
any or all of them produced. 

Take the first case, and suppose the plane cuts Dos half-way 
between Corvus and Cista. Since it does not cut Arctos or Cuspis, 
or either of them produced, it will cut Via, Iter, and Bolus at the 
middle point of each ; and the figure, determined by the inter- 

P 



2IO A New Era of Thought. 

section of the Plane and Mala, is a square. If the length of 
the edge of the cube be taken as the unit, this figure may be 

expressed thus : \ showing that the Z and X lines 

from Corvus are not cut at all, and that the Y line is cut at half- 
a-unit from Corvus. 

Sections taken ^ -^ I ^""^ o o . i ^°^^ ^^° 

be squares. 

Take the second case. 

Let the plane cut Cuspis and Dos, each at half-a-unit from Cor- 
vus, and not cut Arctos or Arctos produced ; it will also cut through 
the middle points of Via and Callis. The figure produced, is a 
rectangle which has two sides of one unit, and the other two are 
each the diagonal of a half-unit squared. 

If the plane cuts Cuspis and Dos, each at one unit from Corvus, 
and is parallel to Arctos, the figure will be a rectangle which has 
two sides of one unit in length ; and the other two the diagonal 
of one unit squared. 

If the plane passes through Mala, cutting £)os produced and 

Cuspis produced, each at one-and-a-half unit from Corvus, and is 

parallel to Arctos, the figure will be a parallelogram like the one 

Z X Y 
obtained by the section q i jl 

This set of figures will be expressed 

ZXY ZXY ZXY 

O . \ . \ O.I.I O.lJ.lJ 

It will be seen that these sections are parallel to each other ; 

and that in each figure Cuspis and Dos are cut at equal distances 

from Corvus. 

We may express the whole set thus : — 

ZXY 

O.I.I 

it being understood that where Roman figures are used, the numbers 

do not refer to the length of unit cut off any given line from Corvus, 

ZXY 
but to the proportion between the lengths. Thus ^ ■, ,t 

means that Arctos is not cut at all, and that Cuspis and Dos are 
cut, Dos being cut twice as far from Corvus as is Cuspis. 

These figures will also be rectangles. 

Take the third case. 



Appendix. 2 1 1 

Suppose Arctos, Cuspis, and Dos are each cut half-way. This 

figure is an equilateral triangle, whose sides are the diagonal of 

Z X Y 
a half-unit squared. The figure is also an equi- 

Z X Y 
lateral triangle, and the figure , . i is an equilateral 

If . If . ig 

hexagon. 

It is easy for us to see what these shapes are, and also, 

Z X Y 
what the figures of any other set would be, as , jj jj 

Z X Y 

or J J J JJ- but we must learn them as a two-dimensional 

being would, so that we may see how to learn the three-dimensional 
sections of a tessaract. 

It is evident that the resulting figures are the same whether we 
fix the cube, and then turn the sectional plane to the required 
position, or whether we fix the sectional plane, and then turn the 
cube. Thus, in the first case we might hav« fixed the plane, and 
then so placed the cube that one plane side coincided with the 
sectional plane, and then have drawn the cube half-way through, in 
a direction at right angles to the plane, when we should have seen 

/Z X Y\ 

the square first mentioned. In the second case ( j j I 

we might have put the cube with Arctos coinciding with the plane 

and with Cuspis and Dos equally inclined to it, and then have 

drawn the cube through the plane at right angles to it until the' 

lines (Cuspis and Dos) were cut at the required distances fromCorvus. 

In the third case we might have put the cube with only Corvus 

coinciding with the plane and with Cuspis, Dos, and Arctos equally 

Z X Y\ 

inclined to it (for any of the shapes in the set j j ,1 

and then have drawn it through as before. The resulting figures 
are exactly the same as those we got before ; but this way is the 
best to use, as it would probably be easier for a two-dimensional 
being to think of a cube passing through his space than to 
imagine his whole space turned round, with regard to the cube. 

We have already seen (p. 117) how a two-dimensional being 
would observe the sections of a cube when it is put with one plane 
side coinciding with his space, and is then drawn partly through. 

Now, suppose the cube put with the line Arctos coinciding with 
his space, and the lines Cuspis and Dos equally inclined to it. At 
first he would only see Arctos, If the cube were moved until 
Dos and Cuspis were each cut half-way, Arctos still being parallel 



212 A Neiv Era of Thought. 

to the plane, Arctos would disappear at once ; and to find out what 
he would see he would have to take the square sections of the cube, 
and find on each of them what lines are given by the new set of 
sections. Thus he would take Moena itself, which may be re- 
garded as the first section of the square set. One point of the 
figure would be the middle point of Cuspis, and since the sectional 
plane is parallel to Arctos, the line of intersection of Moena with 
the sectional plane will be parallel to Arctos. The required line 
then cuts Cuspis half-way, and is parallel to Arctos, therefore it 
cuts Callis half-way. 

Next he would take the square section half-way between Moena 
and Murex. He knows that the line AIvus of this section is 
parallel to Arctos, and that the point Dos at one of its ends is 
half-way between Corvus and Cista, so that this line itself is the 
one he wants (because the sectional plane cuts Dos half-way 
between Corvus and Cista, and is parallel to Arctos). In Fig. 21 
the two lines thus found are shown, a b is the line in Moena, 
and c d the line in the section. He must now find out how far 
apart they are. He knows that from the middle point of Cuspis 
to Corvus is hal-fa-unit, and from the middle point of Dos to 
Corvus is half-a-unit, and Cuspis and Dos are at right angles to 
each other ; therefore from the middle point of Cuspis to the 
middle point of Dos is the diagonal of a square whose sides are 
half-a-unit in length. This diagonal may be written d (J)^. He 
would also see that from the middle point of Callis to the middle 
point of Via is the same length ; therefore the figure is a parallelo- 
gram, having two of its sides, each one unit in length, and the 
other two each d (|)^. 

He could also see that the angles are right, because the lines 

a c and b d are made up of the X and Y directions, and the 

other two, a b and c d, are purely Z, and since they have no tendency 

in common, they are at right angles to each other. 

Z X Y 
If he wanted the figure made by , , it would be a 

little more difficult He would have to take Moena, a section half- 
way between Moena and Murex, Murex and another square which 
he would have to regard as an imaginary section half-a-unit 
further Y than Murex (Fig. 22). He might now draw a ground 
plan of the sections ; that is, he would draw Syce, and produce 
Cuspis and Dos half-a-unit beyond Nugas and Cista. He would 
see that Cadus and Bolus would be cut half-way, so that in the 



Ct\ 


Us 




M.l 


< 


na ^ 




g Stction 1 
< i 


c,^ 


pis 




Syce 



f,g 21. 






i 


Mu 


rex 



Sechon 
kalfxdy 






tZ 


\ Imagnary 
Section 




2. 


\Tofacep. 212 



/."JogSfSKY.. 



cias 



Cuspis 



Ground-plan oF Seetionr 
sAcun in /igi 2 2 . 

■F,g.23. 






F.g.24. 




\Toface #. 213. 



Appendix. 213 

half-way section he would have the point a (Fig. 23), and in Murex 
the point c. In the imaginary section he would have g ; but this 
he might disregard, since the cube goes no further than Murex. 
From the points c and a there would be lines going Z, so that Iter 
and Semita would be cut half-way. 

He could find out how far the two lines a b and^ c d (Fig. 22) 
are apart by referring d and b to Lama, and a and c to Crus. 

In taking the third order of sections, a similar method may be 
followed. 

Suppose the sectional plane to cut Cuspis, Dos, and Arctos, 
each at one unit from Corvus. He would first take Moena, and 
as the sectional plane passes through Ilex and Nugje, the line on 
Moena would be the diagonal passing through these two points 
Then he would take Murex, and he would see that as the plane 
cuts Dos at one unit from Corvus, all he would have is the point 
Cista. So the whole figure is the Ilex to Nugas diagonal, and the 
point Cista. 

Now Cista and Ilex are each one inch from Corvus, and 
measured along lines at right angles to each other ; therefore, they 
are d (i)'' from each other. By referring Nugae and Cista to 
Corvus he would find that they aretelso d (i)^ apart ; therefore the 
figure is an equilateral triangle, whose sides are each d (i)'. 

Suppose the sectional plane to pass through Mala, cutting Cuspis, 
Dos, and Arctos each at \\ unit from Corvus. To find the figure, 
the plane-being would have to take Moena, a section half-way 
between Moena and Murex, Murex, and an imaginary section half-a- 
unit beyond Murex (Fig. 24). He would produce Arctos and Cuspis 
to points half-a-unit from Ilex and Nugas, and by joining these 
points, he would see that the line passes through the middle points 
of Callis and Far (a, b, Fig. 24). In the last square, the imaginary 
section, there would be the point m; for this is i^ unit from 
Corvus measured along Dos produced. There would also be lines 
in the other two squares, the section and Murex, and to find these 
he would have to make many observations. He found the points 
a and b (Fig 24) by drawing a line from r to s, r and s being each 
\\ unit from Corvus, and simply seeing that it cut Callis and Far 
at the middle point of each. He might now imagine a cube Mala 
turned about Arctos, so that Alvus came into his plane ; he might 
then produce Arctos and Dos until they were each i^ unit long, 
and join their extremities, when he would see that Via and Bucina 
are each cut'half-way. Again, by turning Syce into his plane, and 



2 14 -^ New Era of Thought. 

producing Dos and Cuspis to points \\ unit from Corvus and 
joining the points, he would see that Bolus and Cadus are cut half- 
way. He has now determined six points on Mala, through which 
the plane passes, and by referring them in pairs to Ilex, Olus, 
Cista, Crus, Nugae, Sors, he would find that each was d (^ from the 
next ; so he would know that the figure is an equilateral hexagon. 
The angles he would not have got in this observation, and they 
might be a serious difficulty to him. It should be observed that 
a similar difficulty does not come to us in our observation of the 
sections of a tessaract : for, if the angles of each side of a solid 
figure are determined, the solid angles are also determined. 

There is another, and in some respects a better, way by which 
he might have found the sides of this figure. If he had noticed 
his plane-space much, he would have found out that, if a line be 
drawn to cut two other lines which meet, the ratio of the parts of 
the two lines cut off by the first line, on the side of the angle, is 
the same for those lines, and any other two that are parallel to 
them. Thus, if a b and a c (Fig 25) meet, making an angle at a, 
and b c crosses them, and also crosses a' b' and a' c', these last 
two being parallel to a b and a c, then a b : a c : : a' b' : a' c'. 

If the plane-being knew thi», he would rightly assume that if 
three lines meet, making a solid angle, and a plane passes through 
them, the ratio of the parts between the plane and the angle is the 
same for those three lines, and for any other three parallel to them. 

In the case we are dealing with he knows that from Ilex to the 
point on Arctos produced where the plane cuts, it is half-a-unit ;: 
and as the Z, X, and Y lines are cut equally from Corvus, he would 
conclude that the X and Y lines are cut the same distance from 
Ilex as the Z line, that is half-a-unit. He knows that the X line 
is cut at \\ units from Corvus ; that is, half-a-unit from Nugas : 
so he would conclude that the Z and Y lines are cut half-a-unit 
from Nugae. He would also see that the Z and X lines from Cista 
are cut at half-a-unit. He has now six points on the cube, the 
middle points of Callis, Via, Bucina, Cadus, Bolus, and Far. 
Now, looking at his square sections, he would see on Moena a 
line going from middle of Far to middle of Callis, that is, a line 
d i\f long. On the section he would see a line from middle of 
Via to middle of Bolus d {\f long, and on Murex he would see a 
line from middle of Cadus to middle of Bucina, d (J)^ long. Of 
these three lines a b, c d, e f, (Fig. 24)— a b and e f are sides, and 
c d is a section of the required figure. He can find the distances 



Appendix. 215 

between a and c by reference to Ilex, between b and d by refer- 
ence to NugEe, between c and e by reference to Olus, and between 
d and f by reference to Crus ; and he will find that these distances 
are each d i^'' 

Thus, he would know that the figure is an equilateral hexagon 
with its sides d (J)^ long, of which two of the opposite points (c and 
d) are d (i)'' apart, and the only figure fulfilling all these conditions 
is an equilateral and equiangular hexagon. 

Enough has been said about sections of a cube, to show how a 

plane-being would find the shapes in any set as in *; ft ^ 

Z X Y 1 . 11 . U 

°'' I . I . II 

He would always have to bear in mind that the ratio of the 
lengths of the Z, X, and Y lines is the same from Corvus to the 
sectional plane as from any other point to the sectional plane. 
Thus, if he were taking a section where the plane cuts Arctos and 
Cuspis at one unit from Corvus and Dos at one-and-a-half, that 
is where the ratio of Z and of X to Y is as two to three, he would 
see that Dos itself is not cut at all ; but from Cista to the point 
on Dos produced is half-a-unit ; therefore from Cista, the Z and X 
lines will be cut at f of ^ unit from Cista. 

It is impossible in writing to show how to make the various 
sections of a tessaract ; and even if it were not so, it would be 
unadvisable ; for the value of doing it is not in seeing the shapes 
themselves, so much as in the concentration of the mind on the 
tessaract involved in the process of finding them out. 

Any one who wishes to make them should go carefully over the 
sections of a cube, not looking at them as he himself can see them, 
or determining them as he, with his three-dimensional conceptions, 
can ; but he must limit his imagination to two dimensions, and 
work through the problems which a plane-being would have to 
work through, although to his higher mind they may be self- 
evident. Thus a three-dimensional being can see at a glance, 
that if a sectional plane passes through a cube at one unit each 
way from Corvus, the resulting figure is an equilateral triangle. 

If he wished to prove it, he would show that the three bounding 
lines are the diagonals of equal squares. This is all a two- 
dimensional being would have to do ; but it is not so evident to 
him that two of the lines are the diagonals of squares. 

Moreover, when the figure is drawn, we can look at it from a 
point outside the plane of the figure, and can thus see it all at 



2i6 A New Era of Thought. 

once ; but he who has to look at it from a point in the plane can 
only see an edge at a time, or he might see two edges in perspec- 
tive together. 

Then there are certain suppositions he has to make. For 
instance, he knows that two points determine a line, and he 
assumes that three points determine a plane, although he cannot 
conceive any other plane than the one in which he exists. We 
assume that four points determine a solid space. Or rather, we 
say that if this supposition, together with certain others of a like 
nature, are true, we can find all the sections of a tessaract, and of 
other four-dimensional figures by an infinite solid. 

When any difficulty arises in taking the sections of a tessaract, 
the surest way of overcoming it is to suppose a similar difficulty 
occurring to a two-dimensional being in taking the sections of a 
cube, and, step by step, to follow the solution he might obtain, and 
then to apply the same or similar principles to the case in point. 

A few figures are given, which, if cut out and folded along the 
lines, will show some of the sections of a tessaract. But the reader 
is earnestly begged not to be content with looking at the shapes 
only. That will teach him nothing about a tessaract, or four- 
dimensional space, and will only tend to produce in his mind a 
feeling that "the fourth dimension " is an unknown and unthink- 
able region, in which any shapes may be right, as given sections 
of its figures, and of which any statement may be true. While, in 
fact, if it is the case that the laws of spaces of two and three 
dimensions may, with truth, be carried on into space of four 
dimensions ; then the little our solidity (hke the flatness of a 
plane-being) will allow us to learn of these shapes and relations, 
is no more a matter of doubt to us than what we learn of two- and 
three-dimensional shapes and relations. 

There are given also sections of an octa-tessaract, and of a 
tetra-tessaract, the equivalents in four-space of an octahedron and 
tetrahedron. 

A tetrahedron may be regarded as a cube with every alternate 
corner cut off. Thus, if Mala have the corner towards Corvus cut 
off as far as the points Ilex, Nugae, Cista, and the corner towards 
Sors cut off as far as Ilex, Nugas, Lama, and the corner towards 
Crus cut off as far as Lama, Nugae, Cista, and the corner towards 
Olus cut off as far as Ilex, Lama, Cista, what is left of the cube is 
a tetrahedron, whose angles are at the points Ilex, Nugas, Cista, 
Lama. In a similar manner, if every alternate corner of a tessaract 




(i) 



ITd/oci/, 2i6, Nos, i. Id vi. 



jippenatx. 2 1 7 

be cut off, the figure that is left is a tetra-tessaract, which is a 
figure bounded by sixteen regular tetrahedrons. 

The octa-tessaract is got by cutting off every comer of the 
tessaract. If every corner of a cube is cut off, the figure left is 
an octa-hedron, whose angles are at the middle points of the sides. 
The angles of the octa-tessaract are at the middle points of its plane 
sides. A careful study of a tetra-hedron and an octa-hedron as 
they are cut out of a cube will be the best preparation for the study 
of these four-dimensional figures. It will be seen that there is 
much to learn of them, as — How many planes and lines there are 
in each, How many solid sides there are round a point in each. 

A Description of Figures 26 to 41. 

Z 

26 is a section taken i 

27 li 

28 2 

Z 
•^^ ^ 29 is a section taken i 

I ] 30 \\ 

C31 2 

32 2j 

The above are sections of a tessaract. Figures 33 to 35 are of 
a tetra-tessaract. The tetra-tessaract is supposed to be imbedded 
in a tessaract, and the sections are taken through it, cutting the Z 
X and Y lines equally, and corresponding to the figures given of 
the sections of the tessaract. 

Figures 36, 37, and 38 are similar sections of an octa-tessaract. 
Figures 39, 40, and 41 are the following sections of a tessaract. 

Z X Y W 
Z X Y W f ^^ '^ * section taken o . ^ . ^ . ^ 



z 


X 


Y 


W 


I . 


I . 


I . 


I 


z 


X 


Y 


w 


II 


. II 


. II 


. I 



X 


Y 


w 


I 


. I 


I 


4 ■ 


li . 


14 


2 . 


2 . 


2 


X 


Y 


w 


I . 


I 


. \ 


li • 


li • 


i 


2 . 


2 


I 


2i . 


2i . 


li 



It is clear that there are four orders of sections of every four- 
dimensional figure ; namely, those beginning with a solid, those 
beginning with a plane, those beginning with a line, and those 
beginning with a point. There should be little difficulty in finding 
them, if the sections of a cube with a tetra-hedron, or an octa- 
hedron enclosed in it, are carefully examined. 

Q 



PART II. APPENDIX K. 



Model i. MALA. 




Colours : Mala, Light-buff. 

Points: Corvus, Gold. Nugse, Fawn. Cms, Terra-cotta. Cista, Buff. 

Ilex, Light-blue. Sors, Dull-purple. Lama, Deep-blue. Olus, 

Red. 
Lines : Cuspis, Orange. Bolus, Crimson. Cadus, Green-grey. Dos, Blue. 

Arctos, Brown. Far, French-grey. Daps, Dark-slate. Bucina, 

Green. Callis, Reddish. Iter, Bright-blue. Semita, Leaden. 

Via, Deep-yellow. 
Surfaces : Moena, Dark-blue. Proes, Blue-green. Murex, Light -yellow. 

Alvus, Vermilion. Mel, White. Syce, Black. 



PART II. APPENDIX K. 



Model 2. MARGO. 



<t\^* 



6f 





"1 Livflf 






.»^^* 




v,jJ»*^i L.r„ 


y 






l--''^ IThoU* 




>" 






t 

a.' 




V 


4 








.»" 


:5: 




Slilex 




} 


^ 

^-^ 


» 







/ 





Lu<a 



\ 



Colours : Marco, Sage-green. 

Points: Spira, Silver. Ancilla, Turquoise. Mugil, Earthen. Panax, 
Blue-tint. Felis, Quaker-green. Passer, Peacock-blue. Talus, 
Orange-vermilion. Solia, Purple. 

Lines : Luca, Leaf-green. Mappa, Dull-green. Mensura, Dark-purple. 
Opex, Purple-brown. Pagus, Dull-blue. Onager, Dark-pink. 
Vena, Pale-pink. Lixa, Indigo. Tholus, Brown-green. Calor, 
Dark-green. Livor, Pale-yellow. Lensa, Dark. 

Surfaces: Silex, Burnt-sienna. Sal, Yellow-ochre. Povtica, Dun. Crux, 
Indian-red. Lares, Light-grey, Lappa, Bright-green. 



PART II. APPENDIX K. 



Model 3. LAR. 



CO^' 



„os 



O 




%< 



Ca.du 



Syee 



< 



otus 



%t...^ 



Of 



A'' 



appc 



>">? 



^* 



^v^" 



/i 



Luca. 



Colours : Lae, Brick-red. 

Points : Spira, Silver. Ancilla, ■ Turquoise. Mugil, Earthen. Panax, 
Blue-tint. Corvus, Gold. Nugse, Fawn. Crus, Terra-cotta. 
Cista, Buff. 

Lines : Luca, Leaf-green. Mappa, Dull-green. Mensura, Dark-purple. 
Opex, Purple-brown. Ops, Stone. Limus, Smoke. Offex, 
Magenta. Lis, Light-green. Cuspis, Orange. Bolus, Crim 
son. Cadus, Green-grey. Dos, Blue. 

Surfaces: Lotus, Azure. 011a, Rose. Lorica, Sea-blue. Lua, Bright- 
brown. .Syce, Black. Lappa, Bright-green. 



223 



PART II. APPENDIX K. 



Model 4. VELUM. 



\V* 




<i' 



Smmita, 



Cr" 






•^. 



Mel 



Callu 



JVianjo 



ilu 



J' 



'?(■ 



V 



/ 






Tliolu* 



i*' 



m 



'■'*.. 



Tilos 



Colours : Velum, Chocolate. 

Points : Felis, Quaker-green. Passer, Peacock-blue. Talus, Qrange- 
vermilion. Solia, Purple. Ilex, Light-blue. Sors, Dull- 
purple. Lama, Deep-blue. Olus, Red. 

Lines : Tholus, Brown-green. Calor, Dark-green. Livor, Pale-yellow. 
Lensa, Dark. Lucta, Rich-red. Pator, Green-blue. Libera, 
Sea-green. Orsa, Emerald. Callis, Reddish. Iter, Bright- 
blue. Semita, Leaden, Via, Deep-yellow. 

Sttrfaces : Limbus, Ochre. Meatus, Deep-brown. Mango, Deep-green, 
Croeta, Light-red. Mel, White. Lares, Light-grey. 



225 



PART II, APPENDIX K. 



Model s- VESPER. 




Colours : Vesper, Pale-green. 

Points : Spira, Silver. Corvus, Gold. Cista, Buff. Panax, Blue-tint. 

Felis, Quaker-green. Ilex, Light-blue. Olus, Red. Solia, 

Purple. 
Lines : Ops, Stone. Dos, Blue. Lis, Light-green. Opex, Purple-brown. 

Pagus, Dull-blue. Arctos, Brown. Bucina, Green. Lixa, 

Indigo. Lucta, Rich-red. Via, Deep-yellow. Orsa, Emerald. 

Lensa, Dark. 
Surfaces : P^ina, Yellow. Alvus, Vermilion. Camoena, Deep-crimson. 

Crux, Indian-red. Croeta, Light-red. Lua, Light-brown. 



227 



PART II. APPENDIX K. 



Model 6. IDUS. 



*4 



V 



Libera 



.^ 





^ 


Meatus 


,/ 






J^ 


Fator 






f* 










5^ Pa 


ctum V 
i2 


r 




%- 


^ 








s 


3 -^^ 9//fSi 




0''' 


IS 
Q 


Ay oiu 


i 






/ 


i-* 


Limus 


'%>. 








•^fe 





Colours : Idus, Oak. 

Points: Ancilla, Turquoise. Nugse, Fawn. Cras, Terra-cotta. Mugil, 
Earthen. Passer, Peacock-blue. Sors, Dull-purple. Lama, 
Deep-blue. Talus, Orange-vermilion. 

Lines: Limus, Smoke. Bolus, Crimson. Offex, Magenta. Mappa, 
Dull-green. Onager, Dark-pink. Far, French-grey. Daps, 
. Dark-slate. Vena, Pale-pink. Pator, Green-blue. Iter, Bright- 
blue. Libera, Sea-green. Calor, Dark-green. 

Surfaces: Pactum, Yellow-green. Proes, Blue-green. Orca, Dark-grey. 
Sal, Yellow-ochre. Meatus, Deep-brown. Olla, Rose. 



?29 



PART ir. APPENDIX K. 



Model 7. PLUVIUM. 



VeV'* 



•%_ 



C«.Hi« 



^' 



.cXV 



Limluf 
Thai OS 



p»?' 



^t*^■ 



J^oena- 
SiltX 



b! 



■\^ CV/T"-'" • S 



*»l. 



0^'' Z.<7/«J 



^^'.^ 



.cV 



^0^*" 






./ 



Luca 



v^ 



Colours : Pluvium, Dark-stone. 

Points: Spira, Silver. Ancilla, Turquoise. Nugse, Fawn. Corvus, 
Gold. Felis, Quaker-green. Passer, Peacock-blue. Sors, 
Dull-purple. Ilex, Light-blue. 

Lines : Luca, Leaf-green. Limus, Smoke. Cuspis, Orange. Ops, 
Stone. Pagus, Dull-blue. Onager, Dark-pink. Far, French- 
grey. Arctos, Brown. Tholos, Brown-green. Pator, Green- 
blue. Callis, Reddish. Lucta, Rich-red. 

Surfaces: Silex, Burnt-Sienna. Pactum, Yellow-green. Moena, Dark-^ 
blue. Pagina, Yellow. Limbus, Ochre. Lotus, Azure. 



331 



PART II. APPENDIX K. 



Model 8. TELA. 




Colours : Tela, Salmon. 

Points: Panax, Blue-tint. Mugil, Earthen. Crus, Terra-cotta. Cista, 
Buff. Solia, Purple. Talus, Orange-vermilion. Lama, 
Deep-blue. Olus, Red. 

Lines: Mensura, Dark-purple. Offex, Magenta. Cadus, Green-grey. 
Lis, Light-green. Lixa, Indigo. Vena, Pale-pink. Daps, 
Dark-slate. Bucina, Green. Livor, Pale-yellow. Libera, 
Sea-green. Semita, Leaden. Orsa, Emerald. 

Surfaces: Portica, Dun. Orca, Dark-grey. Murex, Light-yellow. 
Camoena, Deep-crimson. Mango, Deep-green. Lorica, Sea- 
blue. 



213 



PART II, APPENDIX K. 



Model 9. SECTION BETWEEN MALA AND MARGO. 



\.Atr>- 



\,u<i' 




Colours : Interior or Tessaract, Wood. 

"Points {Lines) : Ops, Stone. Limus, Smoke. Offex, Magenta. Lis, 

Light-green. Lucta, Rich-red. Pator, Green-blue. Libera, 

Sea-green. Orsa, Emeral d. 
Zines (Surfaces) : Lotus, Azure. OUa, Rose. Lorica, Sea-blue. Lua 

Bright-brown. Pagina, Yellow. Pactum, Yellow-green. 

Orca, Dark -grey. Camoena, Deep-crimson. Limbus, Ochre. 

Meatus, Deep-brown. Mango, Deep-green. Croeta, Light 

red. 
Surfaces (Solids) : Pluvium, Dark-stone. Idus, Oak. Tela, Salmon. 

Vesper, Pale-green. Velum, Chocolate. Lar, Brick-red. 



335 



PART II. APPENDIX K. 



Model io. SECTION BETWEEN LAR AND VELUM. 

Murtx 




>A0 



Mala 
£l Motna 



►s. 



Trio, 



<■■'• 






^' 



Jlla 



rgo. 




V 



f^ 



SiUx 



°-». 



's?;, 



Colours : Interior or Tessaract, Wood. 

Points {Lines) : Pagus, Dull-blue. Onager, Dark-pink. Vena, Pale-pink. 
Lixa, Indigo. Arctos, Brown. Far, French-grey. Daps, 
Dark-slate. Bucina, Green, 
i Lines (Surfaces) : Silex, Burnt-sienna. Sal, Yellow-ochre. Portica, 
Dun. Crux, Indian-red. Pagina, Yellow. Pactum, Yellow- 
green. Orca, Dark-grey. Camoena, Deep-crimson. Moena, 
Dark-blue. Proes, Blue-green. Murex, Light-yellow. Alvus, 
Vermilion. 

Surfacts (Solids) : Pluvium, Dark-stone. Idus, Oak. Tela, Salmon. 
Vesper, Pale-green. Mala, Light-buff. Margo, Sage-green. 



PART II. APPENDIX K. 



Model ii. SECTION BETWEEN VESPER AND IDUS. 



<.. 



■^' 





"j- 


Manffo 








Velora 
Lim-bus 


/ 


5. 


. Pliuv 

/ 

/ 


lum 

/iSCiV 




<n 


Z^ctr 





>^-^ 



Lotus 



A. 



Colours : Interior or Tessaract, Wood. 

Points (Lines) : Luca, Leaf-green. Cuspis, Orange. Cadus, Green-grey. 
Mensura, Dark-purple. Tholus, Brown-green. Callis, Red- 
dish. Semita, Leaden. Livor, Pale-yellow. 

Lines {Surfaces) : Lotus, Azure. Syce, Black. Lorica, Sea-blue. Lap- 
pa, Bright-green. Silex, Burnt-sienna. Moena, Dark-blue. 
Murex, Light-yellow. Portica, Dun. Limbus, Ochre. Mel, 
White. Mango, Deep-green. Lares, Light-grey. 

Surfaces (Solids) : Pluvium, Dark-stone. Mala, Light-buif, Tela, Sal- 
mon. Margo, Sage-green. Velum, Chocolate. Lar, Brick- 
red, 



yg "V; 



PART II. APPENDIX K. 



Model 12. SECTION BETWEEN PLUVIUM AND TELA, 




Colours : Interior or Tessaract, Wood. 

Points {Lines) : Opex, Purple-brown. Mappa, Dull-green. Bolus, 
Crimson. Dos, Blue. Lensa, Dark. Calor, Dark-green. 
Iter, Bright-blue. Via, Deep-yellow. 

Lines {Surfaces) : Lappa, Bright-green. 011a, Rose. Syce, Black. Lua, 
Bright-brown. Crux, Indian-red. Sal, Yellow-ochre. Proes, 
Blue-green. Alvus, Vermilion. Lares, Light-grey. Meatus, 
Deep-brown. Mel, White. Croeta, Light-red. 

Surfaces {Solids) : Margo, Sage-green. Idus, Oak. Mala, Light-buff. 
Vesper, Pale-green. Velum, Chocolate. Lar, Brick-red. 



841